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Quantifying Particle Departure from Axisymmetry in Multiphase Cylindrical Detonation

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Title:
Quantifying Particle Departure from Axisymmetry in Multiphase Cylindrical Detonation
Creator:
Fernandez, Maria Giselle
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
Physical Description:
1 online resource (205 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering
Mechanical and Aerospace Engineering
Committee Chair:
HAFTKA,RAPHAEL TUVIA
Committee Co-Chair:
BALACHANDAR,SIVARAMAKRISHNAN
Committee Members:
KIM,NAM HO
WU,CHANG-YU
MOUSSEAU,VINCENT

Subjects

Subjects / Keywords:
bundled -- cylindrical -- dispersal -- explosive -- instability -- models -- multi-fidelity -- multiphase -- parametric -- simulations -- surrogate -- symmetries
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Aerospace Engineering thesis, Ph.D.

Notes

Abstract:
Dense layers of solid particles surrounding a high-energy explosive generate instabilities after detonation. Conjectures as to the cause of these instabilities include imperfections in the casing, inhomogeneities in the initial distribution of particles, characteristics of the particles, and others. In particular, I study a multiphase detonation with a cylindrical configuration where the initial distribution of particles is initially highly axisymmetric. I do not intend to identify the main physical mechanism responsible for the instabilities observed in the experiments. Alternatively, I assume that it is due to the initial distribution of particles. Therefore, in our simulations, I perturb the particle volume with azimuthal sinusoidal waves and I study the final distribution of the particles quantifying the amplification of the departure from axisymmetry. I observe an instability mechanism, that is possibly partially the result of non-classical Raleigh-Taylor and/or Richmyer-Meshkov instabilities. I have called it channeling instability and it has two main effects (i) accelerate particles located in low particle volume radial sectors and (ii) push particles from low particle volume sectors to high particle volume sectors. In other words, the channeling effect increases the angular variation in the net volume of particles contained within radial sectors of the domain and also increases the difference in the radial extent of particle distribution between the different sectors. To quantify the particles departure from axisymmetry, one L2 metric based on energy and another Linfinity metric based on the maximum difference in particle volume between radial sectors have been constructed. The variables considered are the parameters of a trimodal sinusoidal perturbation (amplitudes, wavelengths, and relative phases). I found that the metric dependence on the relative phases is negligible and that unimodal perturbations amplify both metrics the most. Our simulations are expensive and surrogate models can achieve accurate predictions of the dependence of the metrics on amplitudes and wavelengths at low computational cost. This work includes a substantial literature review of the design of experiments, surrogate models and multi-fidelity surrogate models. Using a large number of simulations, I construct low-fidelity, high-fidelity and multi-fidelity surrogate models using an additive, multiplicative, and comprehensive corrections. I use linear regression which basis functions are monomials up to a quadratic polynomial. I find that, for this problem with this particular surrogate and basis functions, multi-fidelity models have better performance compared with single-fidelity models. This is due to the complex behavior of both LF and HF functions, but a high correlation between them that allows correcting the LF function with a low-order polynomial. Due to the symmetrical features of our problem, I explore options for reducing the number of simulation used to construct surrogates while maintaining accuracy by taking advantage of parametric symmetries. These symmetries allow us to obtain free information and, therefore, the possibility of cheaper or more accurate predictions. I impose the inherent parametric symmetries of our model while building the multi-fidelity surrogate and I compare the performance with the multi-fidelity surrogate without imposing symmetries. I find that for a small number of high-fidelity points the performance of the surrogate using symmetries is much better, however, for more than 100 HF data points their performance is indistinguishable. This is due to the use of quadratic polynomial surrogates, which do not benefit from additional data when the number of data points in much larger than the number of coefficients. This work includes three publications (Chapter 3, Chapter 4, and Chapter 5), which have been included without modifications as they were published or sent to the journal. Therefore, I warn the reader that there are some repetitions in the introduction and in the final chapters. Also, there are small discrepancies in notation between chapters, however, the notation is clearly stated at the beginning of each one. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2018.
Local:
Adviser: HAFTKA,RAPHAEL TUVIA.
Local:
Co-adviser: BALACHANDAR,SIVARAMAKRISHNAN.
Statement of Responsibility:
by Maria Giselle Fernandez.

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UFRGP
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Applicable rights reserved.
Classification:
LD1780 2018 ( lcc )

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QUANTIFYINGPARTICLEDEPARTUREFROMAXISYMMETRYINMULTIPHASECYLINDRICALDETONATIONByMARIAGISELLEFERNANDEZADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2018

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c2018MaraGiselleFernandez

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Amisdosanglesdelaguarda,eldelcieloyeldelatierra

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ACKNOWLEDGMENTSMegustaradedicarestetrabajoamifamilia,quemeapoyaaladistancia.Amita,PochiGodino,quesiempreestuvopresentetomandoelpapeldeunamadre.Amipadre,ElsoHugoFernandez,quenuncadejodecreerenm.ADominiqueFratantonioqueiluminomisultimasetapasdemisestudiosdedoctorado.AFrederickOuelletquemeayudomuchoconlagramticayconcafe.Atodosmisamigosporsualiento.Finalmente,megustaraagradecerespecialmenteamisdirectores,Dr.RaphaelT.HaftkayDr.S.Balachandar,porsugranayudaenesteproceso,brindandomesusmejoresconsejos.Idedicatethisworktomyfamilywhichsupportsmefromthedistance.Tomyaunt,PochiGodino,whowasalwayspresenttakingtheroleofamother.Tomydad,ElsoHugoFernandez,whohadneverstoppedbelievinginme.ToDominiqueFratantoniothatbrightenedupmylaststagesofthePd.D.ToFrederickOuelletwhohelpedmealotwithgrammarandcoee.Toallmyfriendsfortheirencouragement.Finally,Iwouldliketogivespecialthankstomyadvisors,Dr.RaphaelT.HaftkayDr.S.Balachandar,fortheirgreathelpinthisprocess,givingmetheirbestadvice.Dedicoquestolavoroallamiafamigliachemisostieneadistanza.Amiazia,Pochicheesemprestatapresentenelruolodiunamadre.Amiopadre,ElsoHugoFernandez,chenonhamaismessodicredereinme.ADominiqueFratantoniochehailluminatolemieultimefasideldottorato.AFrederickOuelletchemihaaiutatomoltoconlagrammaticaeconilca.Atuttiimieiamiciperilloroincoraggiamento.Inne,vorreiringraziareinmodoparticolareimieisupervisori,Dr.RaphaelT.HaftkayDr.S.Balachandar,perillorograndeaiutoinquestoprocesso,dandomiilloromiglioriconsigli.ThisworkissupportedbytheCenterforCompressibleMultiphaseTurbulence,theU.S.DepartmentofEnergy,NationalNuclearSecurityAdministration,AdvancedSimulationandComputingProgram,asaCooperativeAgreementunderthePredictiveScienceAcademicAllianceProgram,underContractNo.DE-NA0002378. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 14 CHAPTER 1INTRODUCTION .................................. 17 1.1Motivation .................................... 17 1.2Multi-delitySurrogatesandSymmetries ................... 19 2LITERATUREREVIEW .............................. 22 2.1MutiphaseDetonationandNon-classicalInstabilities ............ 22 2.2StrategiesforDesignofExperimentinMulti-delitySurrogateModels .. 23 2.3SurrogateModels ................................ 30 2.4SomeStatisticsaboutMulti-delityModels ................. 32 2.4.1TypesofFidelity ............................ 36 2.4.2MethodsforCombiningFidelities ................... 41 2.4.2.1Multi-delitysurrogatemodelsvs.multi-delityhierarchicalmodels ............................. 41 2.4.2.2Multi-delitysurrogatemodels ............... 42 2.4.3DeterministicMethods(DM)vs.Non-deterministicMethods(NDM) 45 3ISSUESINDECIDINGWHETHERUSINGMULTI-FIDELITYSURROGATES 51 3.1Summary .................................... 51 3.2Nomenclature .................................. 51 3.3Background ................................... 52 3.4ProgressinFittingSurrogatestotheData .................. 53 3.5CostRatiovs.Savings ............................. 57 3.6ChallengesinFittingMulti-delitySurrogates ................ 61 3.6.1DecidingwhethertoUsetheLow-delityData ............ 61 3.6.2ChoosingbetweenMultipleLow-delityDatasets ........... 63 3.6.3SelectingforOtherSurrogates .................... 63 3.7Recommendations ................................ 64 3.8ConcludingRemarks .............................. 66 4EARLYTIMEEVOLUTIONOFCIRCUMFERENTIALPERTURBATIONOFINITIALPARTICLEVOLUMEFRACTIONINEXPLOSIVECYLINDRICALMULTIPHASEDISPERSION ............................ 67 5

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4.1Summary .................................... 67 4.2Background ................................... 68 4.3NumericalMethods ............................... 72 4.3.1ComputationalModeling ........................ 72 4.3.1.1Governingequations ..................... 72 4.3.1.2Physicalmodels ........................ 74 4.3.2ProblemDescription .......................... 77 4.4PerturbationMethodology ........................... 80 4.5CasesSimulated ................................. 83 4.6Results:InstabilityCharacterization ..................... 86 4.6.1UnperturbedCase ............................ 86 4.6.2Perturbedcases ............................. 89 4.6.3ChannelingEect ............................ 92 4.6.4ParticleLocation ............................ 97 4.6.5ShockandParticleBedEvolution ................... 103 4.7Results:ComparisonofInstability ....................... 103 4.7.1TheNormalizedFourierEectiveVariation .............. 103 4.7.2TimeEvolutionandStatisticalNoise ................. 108 4.7.3TheWeakDependenceonPhase .................... 111 4.7.4TheStrongDependenceontheEectiveWavenumber ........ 112 4.7.5TheDependenceontheModeAmplitude ............... 114 4.7.6TheNormalizedMaximumParticleVolumeFractionDierence ... 114 4.8ConcludingRemarks .............................. 117 5ONTHEUSEOFSYMMETRIESINBUILDINGSURROGATEMODELS .. 119 5.1Summary .................................... 119 5.2Nomenclature .................................. 119 5.3Background ................................... 121 5.4ApplyingSymmetries .............................. 123 5.5SurrogateImplementation ........................... 128 5.5.1Kriging .................................. 128 5.5.2LinearRegression ............................ 130 5.6AnalyticalExample ............................... 131 5.6.1ProblemDescription .......................... 131 5.6.2DesignofExperiments ......................... 132 5.6.3KrigingResults ............................. 132 5.6.4LinearRegressionResults ........................ 135 5.7PhysicalExample ................................ 141 5.7.1ProblemDescription .......................... 141 5.7.2DesignofExperiments ......................... 144 5.7.3KrigingResults ............................. 146 5.7.4LinearRegressionResults ........................ 151 5.8ConcludingRemarks .............................. 153 5.9Data ....................................... 155 6

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6METRICSTUDY .................................. 157 6.1TheNormalizedFourierEectivePerturbation ................ 157 6.2TheUnlteredNormalizedMaximumParticleVolumeDierence ...... 159 6.3MetricsComparison .............................. 159 7MULTI-FIDELITYSURROGATES ......................... 166 7.1TheMulti-delitySurrogatesUsed ...................... 166 7.2TheDesignofExperiments ........................... 169 7.3TheSurrogatePerformance .......................... 170 7.4UsingSymmetriesintheConstructionofMulti-delitySurrogates ..... 176 8CONCLUSIONS ................................... 178 REFERENCES ....................................... 181 BIOGRAPHICALSKETCH ................................ 205 7

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LISTOFTABLES Table page 2-1FluidMechanicsorientedpapersperLFmodelandHFmodelused. ....... 38 2-2FluidMechanicsorientedpapersbyLFmodelandHFmodelused. ....... 39 2-3SolidMechanicsorientedpaperspertypeofanalysisusedtodeterminedelity. 39 2-4SolidMechanicsorientedpaperspertypeofdelityusedbesidesanalysistype. 40 2-5Papersthatusedeterministicmethods(DM)fortheconstructionoftheMFsurrogatemodel. ................................... 50 2-6Papersthatusenon-deterministicmethods(NDM)toconstructtheMFsurrogatemodel. ......................................... 50 3-1Padronetal.,2016[1]optimizationcost,savingsandaccuracyreportasamodelforauthors. ...................................... 65 4-1Griddetails ...................................... 79 4-2Casessimulated .................................... 85 5-1Kriginghyperparameters1and2for^y,^yOS,^yRD,and^yAPPoftheanalyticalfunction ........................................ 134 5-2RMSEandDSforKrigingapproximations^y,^yOS,^yRD,and^yAPPoftheanalyticalfunctioncomputedintheregionofinterestD ................... 134 5-3RMSEandDSerrorsforlinearregressionapproximations^y,^yOS,^yRD,^ySBF,and^yAPPoftheanalyticalfunctioncomputedintheregionofinterestD .... 138 5-4Kriginghyperparameters .............................. 148 5-5Krigingsurrogateerrorsfor^ ............................ 148 5-6Kriginghyperparametersfor ............................ 150 5-7RMSEandDScalculatedinthedomainofinterestDfor^,^OS,^RDand^APPusingKrigingsurrogateforthephysicalexample .................. 150 5-8RMSEandDScalculatedinthedomainofinterestDfor ............ 151 5-9RMSEandDScalculatedinthedomainofinterestfor ............. 152 5-10Originalsimulationtrainingdatapointsforsurrogateconstructioninthephysicalexample ........................................ 155 5-11Originalsetofsimulationtestdatapointsusedtotesttheperformanceofthesurrogatemodelsinthephysicalexample ...................... 156 8

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6-1Parametervaluesofthesixcasesstudied.SMstandsforsinglemodal,BMforbimodal,andTMfortrimodal. ........................... 160 7-1Correlationcoecientbetweenthedatapoints ................... 169 7-2RelativeRMSEandcoecientofdetermination(R2)forthemetricFusing711HFdatapoints. ................................. 173 7-3RelativeRMSEandcoecientofdetermination(R2)forthemetricusing711HFdatapoints. .................................... 173 9

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LISTOFFIGURES Figure page 1-1DavidL.Frost.EvolutionintimeofacylindricalmultiphasedetonationusingPETNexplosivesurroundedbyglassparticles.17September2012.Quebec,Canada.Source:D.L.Frost,Y.Gregoire,O.Petel,S.Goroshin,andF.Zhang,Particlejetformationduringexplosivedispersalofsolidparticles,PhysicsofFluids,vol.24,no.9,p.91-109,2012. .............................. 18 2-1LFmodelsarecheaperbecausetheyareusuallyasimplicationofHFmodels. 24 2-2IfMFmodelsinvolvetheconstructionofasurrogatemodeltoexplicitlycombinedelities(e.g.co-Kriging)itiscalledMFsurrogatemodel. ............ 25 2-3FFDandCCDsamplingstrategies. ......................... 27 2-4Nestedsamplingdesign. ............................... 29 2-5Nearestneighborsampling. ............................. 29 2-6ProportionofdierentattributesconsideredintheMFmodelpapersreviewed,thechartsarebasedon178papers. ......................... 34 2-7Maindierencesbetweendelitiesfoundintheliterature. ............. 37 2-8Ofthetotalofthe178papersreviewed,127constructedamulti-delitysurrogatemodeltoexplicitlycombinethedelities.TherestofthepaperspresentanMFmodelusingmulti-delityhierarchicalmodels. ................... 42 2-9Schematicofconstantcorrectionfactors. ...................... 44 2-10MFsurrogatemodelsparameterscanbedeterminedusingdeterministicmethods(DM)ornon-deterministicmethods(NDM). .................... 46 2-11Ofthe127reviewedpapersthatconstructaMFSmodel,54%usedeterministicmethods(DM)while46%usenon-deterministicmethods(NDM). ........ 48 2-12Percentageofpapersreviewedpublishedfromearly90suntilthepresentfordeterministicmethods(DM)andnon-deterministicmethods(NDM). ...... 49 3-1Low-delity,yL(x)),andhigh-delity,yL(x),functions. .............. 55 3-2CostratiobetweenasingleanalysisoftheLFMandasingleanalysisoftheHFMvs.costratiobetweentheoptimizationprocessusinganMFMandtheoptimizationprocessusinganHFM. ................................ 58 3-3Eectofthesurrogatesused.Thereisnoclearrelationshipbetweenthecasesthatusethesamesurrogate. ............................. 59 10

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4-1DavidL.Frost.EvolutionintimeofacylindricalmultiphasedetonationusingPETNexplosivesurroundedbyglassparticles.17September2012.Quebec,Canada.Source:D.L.Frost,Y.Gregoire,O.Petel,S.Goroshin,andF.Zhang,Particlejetformationduringexplosivedispersalofsolidparticles,PhysicsofFluids,vol.24,no.9,p.91-109,2012. .............................. 69 4-2Schematicofthecomputationaldomain(nottoscale). .............. 77 4-3Schematicofthegridsetupusedforillustrationpurposesonly. .......... 79 4-4Simulationmetricsasafunctionoftimeforthethreedierentgrids. ...... 80 4-5ThemetricFasafunctionofkeff.Bluedotsrepresentsthecaseswithatrimodalperturbationwhiletheorangedotsthecaseswithasinglemodalperturbation 81 4-6PVFcontoursattheinitialtime. .......................... 84 4-7Evolutionofgasdensity,PVF,PS,CI,IPF,andOPFfortheunperturbedcase(A1=A2=A3=0). ................................. 88 4-8UnperturbedinitialPVF.Gaspressure,gastemperature,particlevelocity,andPVFcontoursfort=500s. ............................. 89 4-9Evolutionofgasdensity,PVF,PS,CI,IPF,andOPFfortheunimodalperturbedcasewithparametersA1=p 0:02;A2=A3=0;k1=10;1=0. ......... 91 4-10PerturbedinitialPVF. ................................ 92 4-11Gasvelocity(left)andPVFcontours(right)atnaltime. ............ 95 4-12GasvelocityandPVFcontoursforthecaseA1=0:039145,A2=0:116199,A3=0:070466,k1=1,k2=12,k3=8,1=0,2=6:000171,3=0:378179. 96 4-13GasvelocityandPVFcontoursforthecaseA1=0:008722,A2=0:115284,A3=0:081446,k1=14,k2=2,k3=25,1=0,2=3:923277,3=3:135248. 97 4-14EvolutionofthePV,rPV,r2PV,r3PV,andr4PVfortheunimodalcasek=8. 100 4-15EvolutionofthePV,rPV,r2PV,r3PV,andr4PVforthetrimodalcase1. ... 101 4-16EvolutionofthePV,rPV,r2PV,r3PV,andr4PVforthetrimodalcase2. ... 102 4-17ShockandmeanPVlocationasafunctionoftheradius. ............. 104 4-18PVamplitudespectrumsquaredforA1=0:130;A2=0:039;A3=0:039,k1=8;k2=17;k3=15,and1=0:200;2=0:520;3=4:851.ThespectrumisnormalizedbyjA0j2. ................................. 106 4-19Amplitudeevolutionofinitialandemergingmodes. ................ 107 4-20ThemetricFasafunctionoftimeforfourdierentkeff. ............. 109 11

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4-21Variabilityinthemetric. ............................... 112 4-22F(t=500s)asafunctionofkeff. ......................... 113 4-23Positiveoctantoftheamplitudespherewithradiusp 0:02fordierentrangesofkeff.ThecolorbarshowsthemetricF(t=500s)inthatinterval. ...... 115 5-1Schematicofthevelocityeldofalaminarowinasquareduct.Itsatisesreectionalsymmetryaboutthemidplanesandaboutthediagonals,andsatisesrotationalsymmetryto90,180,and270rotations ............... 122 5-2ExampleofaDoEforAPP,OS,andRDapproachesfortheuseofsymmetries.EmptycirclesaretheoriginalDoEofno=3points(N0),crossestheircorrespondentns=3permutationpoints(Ns).Fullcirclesintherightsubguresaretheresultingnudatapointsforeachapproach(Nu) .................. 126 5-3AnalyticalfunctiondescribedbyEq.(5-12). .................... 133 5-4Krigingsurrogateerrors:crossesrepresenttheRMSEforeachoneofthe100DoEsof28pointsoutofthe49.Themeanofthe100DoEsRMSEishighlighted.TheRMSEforeachapproachisalsonoted ..................... 134 5-5ContourplotsoftheKrigingapproximationsAPP,RDandOS. ......... 136 5-6ContourplotsoftheabsoluteerrorcontoursoftheKrigingapproximationsAPP,RD,andOS.ThemaximumabsoluteerrorisobtainedusingOSandrepresentsan8%ofthefunctionrangeinD. .......................... 137 5-7Linearregressionsurrogateerrors:crossesrepresenttheRMSEforeachoneofthe100DoEsof28pointsoutofthe49. ...................... 138 5-8ContourplotsofthelinearregressionapproximationsOS,RD,APPandSBF. 139 5-9ContourplotsofthelinearregressionapproximationsOS,RD,APPandSBF. 140 5-10Schematicofthetwo-dimentionalcomputationaldomain(nottoscale) ..... 142 5-11PVFcontoursatinitialtime. ............................ 143 5-12Setofthenutrainingdatapoints,Nu,usedtotrainthesurrogatesforeachcasetoapproximate. ................................... 146 5-13Setofthenutrainingdatapoints,Nu,usedtotrainthesurrogatesforeachcasetoapproximate. ................................... 147 5-14Krigingsurrogateerrors:crossesrepresenttheRMSEforeachoneofthe100DoEof34datapointsoutof204. .......................... 149 5-15Krigingsurrogateerrorsfor^ ............................ 150 5-16Linearregressionsurrogateerrorsfor^. ....................... 152 12

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5-17Linearregressionsurrogateerrorsfor^. ....................... 153 6-1PVamplitudespectrumsquaredforA1=0:13,A2=0:039,A3=0:039,k1=8,k2=17,k3=15,and1=0:200,2=0:520,3=4:851.ThespectrumisnormalizedbyjA0j2. ................................. 158 6-2PVcontourevolution. ................................ 161 6-3asafunctionoftimeforthesixcasesdescribedinTable6-1. .......... 163 6-4ParticlevolumeasafunctionoftheangularcoordinateforthecaseSM. ... 163 6-5ParticlevolumeasafunctionoftheangularcoordinateforthecaseTM3. ... 164 6-6NormalizedFourierspectrasquarednormalizedforthecaseSM. ......... 165 6-7SquaredFourierspectranormalizedforthecaseTM3. ............... 165 7-1Themetrics,Fand,asafunctionofkeff. .................... 171 7-2RMSEforbothmetrics,Fand,asafunctionofthenumberofHFdatapointsusedtotrainthesurrogates. ............................. 172 7-3TheconstantandthesquaredsumofthecoecientsbiasafunctionoftheHFdatapointsforthecomprehensiveMFcorrectionsshowninFigure7-2. ... 175 7-4MeancontributionoftheLF(^yLF(x)fromEq.(7-8))andoftheHF(Pp1=1Xi(x)bifromEq.(7-8))modelstothecomprehensiveMFsurrogateprediction(usingLFdatapoints)inpercentageasafunctionoftheHFdatapointsused. ..... 176 7-5RMSEforbothmetrics,Fand,asafunctionofthenumberofHFdatapointsusedtotrainthesurrogatesusingandnotusingpermutationpointsfortheMFsurrogate^yMF6. .................................... 177 13

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyQUANTIFYINGPARTICLEDEPARTUREFROMAXISYMMETRYINMULTIPHASECYLINDRICALDETONATIONByMaraGiselleFernandezDecember2018Chair:RaphaelT.HaftkaCochair:S.BalachandarMajor:AerospaceEngineeringDenselayersofsolidparticlessurroundingahigh-energyexplosivegenerateinstabilitiesafterdetonation.Conjecturesastothecauseoftheseinstabilitiesincludeimperfectionsinthecasing,inhomogeneitiesintheinitialdistributionofparticles,characteristicsoftheparticles,andothers.Inparticular,Istudyamultiphasedetonationwithacylindricalcongurationwheretheinitialdistributionofparticlesisinitiallyhighlyaxisymmetric.Idonotintendtoidentifythemainphysicalmechanismresponsiblefortheinstabilitiesobservedintheexperiments.Alternatively,Iassumethatitisduetotheinitialdistributionofparticles.Therefore,inoursimulations,IperturbtheparticlevolumewithazimuthalsinusoidalwavesandIstudythenaldistributionoftheparticlesquantifyingtheamplicationofthedeparturefromaxisymmetry.Iobserveaninstabilitymechanism,thatispossiblypartiallytheresultofnon-classicalRaleigh-Taylorand/orRichmyer-Meshkovinstabilities.Ihavecalleditchannelinginstabilityandithastwomaineects(i)accelerateparticleslocatedinlowparticlevolumeradialsectorsand(ii)pushparticlesfromlowparticlevolumesectorstohighparticlevolumesectors.Inotherwords,thechannelingeectincreasestheangularvariationinthenetvolumeofparticlescontainedwithinradialsectorsofthedomainandalsoincreasesthedierenceintheradialextentofparticledistributionbetweenthedierentsectors. 14

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Toquantifytheparticlesdeparturefromaxisymmetry,oneL2metricbasedonenergyandanotherL1metricbasedonthemaximumdierenceinparticlevolumebetweenradialsectorshavebeenconstructed.Thevariablesconsideredaretheparametersofatrimodalsinusoidalperturbation(amplitudes,wavelengths,andrelativephases).Ifoundthatthemetricdependenceontherelativephasesisnegligibleandthatunimodalperturbationsamplifybothmetricsthemost.Oursimulationsareexpensiveandsurrogatemodelscanachieveaccuratepredictionsofthedependenceofthemetricsonamplitudesandwavelengthsatlowcomputationalcost.Thisworkincludesasubstantialliteraturereviewofthedesignofexperiments,surrogatemodelsandmulti-delitysurrogatemodels.Usingalargenumberofsimulations,Iconstructlow-delity,high-delityandmulti-delitysurrogatemodelsusinganadditive,multiplicative,andcomprehensivecorrections.Iuselinearregressionwhichbasisfunctionsaremonomialsuptoaquadraticpolynomial.Indthat,forthisproblemwiththisparticularsurrogateandbasisfunctions,multi-delitymodelshavebetterperformancecomparedwithsingle-delitymodels.ThisisduetothecomplexbehaviorofbothLFandHFfunctions,butahighcorrelationbetweenthemthatallowscorrectingtheLFfunctionwithalow-orderpolynomial.Duetothesymmetricalfeaturesofourproblem,Iexploreoptionsforreducingthenumberofsimulationusedtoconstructsurrogateswhilemaintainingaccuracybytakingadvantageofparametricsymmetries.Thesesymmetriesallowustoobtainfreeinformationand,therefore,thepossibilityofcheaperormoreaccuratepredictions.Iimposetheinherentparametricsymmetriesofourmodelwhilebuildingthemulti-delitysurrogateandIcomparetheperformancewiththemulti-delitysurrogatewithoutimposingsymmetries.Indthatforasmallnumberofhigh-delitypointstheperformanceofthesurrogateusingsymmetriesismuchbetter,however,formorethan100HFdatapointstheirperformanceisindistinguishable.Thisisduetotheuseofquadraticpolynomial 15

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surrogates,whichdonotbenetfromadditionaldatawhenthenumberofdatapointsinmuchlargerthanthenumberofcoecients.Thisworkincludesthreepublications(Chapter 3 ,Chapter 4 ,andChapter 5 ),whichhavebeenincludedwithoutmodicationsastheywerepublishedorsenttothejournal.Therefore,Iwarnthereaderthattherearesomerepetitionsintheintroductionandinthenalchapters.Also,therearesmalldiscrepanciesinnotationbetweenchapters,however,thenotationisclearlystatedatthebeginningofeachone. 16

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CHAPTER1INTRODUCTION 1.1MotivationExperimentshaveshownthatdenselayersofsolidparticlessurroundingahighenergyexplosiveundergoinstabilitiesastheyradiallydispersefollowingthedetonation[ 2 { 4 ].Figure 1-1 showsacylindricalmultiphasedetonationexperimentperformedbyFrostetal.in2012whichpresentsaninitiallycylindricallysymmetricconguration.Herethecentralcylindricalchargeissurroundedbyanannularbedofnearlysphericalparticles.Atlatertimes,however,thissymmetryislostduetothedevelopmentofinstabilities.Themechanismsgoverningtheformationandgrowthoftheseparticleinstabilitiesarepoorlyknown,butmightdependonthenatureoftheparticles,thegeometryofthecharge,themassratioofexplosivetoparticles,imperfectionsinthecasingcontainingtheparticles,inhomogeneitiesintheinitialdistributionofparticles,stresschainswithintheparticlebedduringshockpropagation,andothercausesnotyetconsidered.However,themainsourceoftheinstabilitiesandtheirgrowth/amplicationmechanismsarestillunknown.Thepossiblesourcesaremanyandthemechanismsofgrowthareoftencomplexandinteracting.Therefore,thefocusofthisworkislimitedtoinvestigatingafewspecicaspectsregardingthegrowthofdeparturesfromaxisymmetryobservedintheexperiments.Forinstance,inthiswork,Iassumethesourceoftheinitialperturbationtobeanon-axisymmetricparticledistributionanddonotexploreotherpossibilities.Ourfocuswillbeoncharacterizinghowthenatureoftheinitialperturbationinuencestheinstabilitygrowthatlatertimes.Previousresearch[ 5 { 7 ]indicatesthatunimodalandbimodalazimuthalperturbationsoftheparticlevolumefractionleaveasignatureintheparticlecloudfortimesontheorderofmillisecondsandmoreafterdetonation.Inthiswork,Ihavechosentoimposeuptotrimodalperturbationsintheinitialparticlevolumefraction.Withanensembleofmorethan1,800simulationsthatcoverawiderangeofinitialperturbations,Istudythe 17

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A B CFigure1-1. DavidL.Frost.EvolutionintimeofacylindricalmultiphasedetonationusingPETNexplosivesurroundedbyglassparticles.17September2012.Quebec,Canada.Source:D.L.Frost,Y.Gregoire,O.Petel,S.Goroshin,andF.Zhang,Particlejetformationduringexplosivedispersalofsolidparticles,PhysicsofFluids,vol.24,no.9,p.91-109,2012.A)Initialtime,beforethedetonation.Thecongurationishighlyaxisymmetric;B)2:5mstimeafterdetonation.Instabilitiesbegintoformmakingthecongurationrelativelyaxisymmetric;C)5msafterthedetonation.Instabilitiesarehighlydevelopedandthedeparturefromaxisymmetryisevident. amplicationofdeparturefromaxisymmetryatlatertimesanditsdependenceonthenatureoftheinitialperturbation.Thefocusofthisdissertationistoconsideraverylargenumberofnumericalsimulationswheretheinitialmodalperturbationoftheparticledistributionwithintheannularbedissystematicallyvaried.Inordertocomparehowtheradiallyexpandingparticledispersiondepartsfromanaxisymmetricdistribution,Irstidentifymetricsthatwillbeshowntoperforminarobustmanner.Twometricswerechosen,whichrepresentformsofL2andL1norms.Duetotherandomnatureoftheparticledistribution,thereisanunavoidablestochasticvariationinquantitiessuchasparticlevolumefraction.Forexample,twodierentsimulationswithidenticalkeyparametersthataredierentonlyintherandomlocationoftheinitialparticledistribution,areshowntoevolvedierently.Suchsimulationsareusedtoestablishtherandomnoiselevel.Thenthekeyparametersoftheinitialperturbationwhoseinuenceismuchstrongerthanthenoiselevel,areidentied.Iobserveaninterestingchannelinginstabilitythatcontributestothegrowthof 18

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departurefromaxisymmetry.ThisinstabilityisrelatedtoRTinstability,butoneinwhichtheentirelayerofparticlesseemstoparticipate,notjustthefront. 1.2Multi-delitySurrogatesandSymmetriesDespitegreatstridesincomputationalpoweroverthepastfewdecades,investigationofcomplexproblemsthroughcomputationalsimulationsremainsachallenge.Therefore,whensimulationsarecomputationallyexpensiveandmultiplerealizationsareneeded,asinuncertaintyquantication(UQ)[ 8 { 10 ],inverseproblems[ 11 { 13 ]oroptimization[ 14 { 16 ],surrogatemodelsbecomeanattractiveoption.Mostsurrogatesarealgebraicmodelsthatapproximatetheresponseofasystembasedonttingalimitedsetofcomputationallyexpensivesimulationsinordertopredictaquantityofinterest.Somesurrogatescombinemultiplemodeldelities[ 17 18 ].Theiraccuracyisinuencedbythedesignofexperiments(DoE)used,thesizeofthedomainofinterest,thesimulationaccuracyatthedatapointsandthenumberofsamplesavailablefortheirconstruction[ 19 ].High-delity(HF)modelsusuallyrepresentthebehaviorofasystemtoacceptableaccuracyfortheapplicationintended.Thesemodelsareusuallyexpensiveandtheirmultiplerealizationsoftencannotbeaorded.Ontheotherhand,low-delity(LF)modelsarecheaperbutlessaccurate.Theyareobtainedbydimensionalityreduction[ 20 ],simplerphysicsmodels[ 21 ],coarserdiscretization[ 22 ],partiallyconvergedresults[ 23 ],etc.Multi-delity(MF)modelscombinetheinformationofboth,LFandHF,andhavedrawnmuchattentioninthelasttwodecadesbecausetheyholdthepromiseofachievingthedesiredaccuracyatlowercost.Surrogatesareoftenusedforapproximationscreatedtoreducecomputationalcostwhenalargenumberofexpensivesimulationsareneededforsuchprocessesasoptimization[ 14 16 ]anduncertaintyquantication(UQ)[ 24 ].Surrogatescanbeconstructedtoreducethecostofsingledelitymodels.Surrogatemodelsconstructedusinginformationfrommodelsofdierentdelitiesareusuallyknownasmulti-delity 19

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(MF)surrogates.IapproximatetheL2andL1metricsusingMFsurrogates,constructedusingtwolevelsofdelities.Herethelevelofdelityisdictatedbythegriddiscretization.TheMFsurrogateperformancewascomparedwiththesingle-delityperformance.Symmetrieshaveplayedanimportantroleinthemodelingofcomplexprocesses.Symmetriesinherenttoaproblemmightleadtocostsavingsinsurrogateconstruction.Mathematically,manydierentsymmetriescanbedened.Themostcommonaresymmetriesbasedongeometry,suchasreectionalsymmetry,rotationalsymmetry,translationalsymmetryandsoon.Thesesymmetriescanalsobeconsideredintheparametricspace.Letusconsideradvariablefunctionf(x),wherex=(x1;x2;x3;:::;xd),suchthattheorderofthevariablesdoesnotmatter,i.e.f(x1;x2;x3;:::;xd)=f(x2;x1;x3:::;xd)=f(x3;x1;x2:::;xd)andsoon.Therefore,thefunctiontakesthesamevalueatd!points(thedatapointanditspermutations).Thesepermutationsoftheoriginaldatapointwillbecalledpermutationspoints.Ifweallowrepetitionsofthevariablevaluesinthesamedatapoint,thenumberofpermutationpointswouldbeless.Forexample,ifthevalueofvariablex1andvariablex2arethesame,thenthefunctionwilltakethesamevalueatd!=2!points.Thisnumberwilldecreaseevenmoreifthenumberofrepetitionsincreases.Theperturbationsimposedintheparticlevolumefractionpresentthiskindofsymmetriesduetothefactthattheorderofthethreepermutationmodesdoesnotmatter.SymmetriesinMFsurrogateswereimposedlookingforfurthercostreductioninsimulationsoraccuracyimprovementsinsurrogates.Thisdissertationisorganizedasfollows.InChapter 2 ,Ireviewtheliteratureofthreeimportanttopicsinourwork:instabilitiesinmultiphasedetonations,strategiesforthedesignofexperimentsinMFsurrogates,surrogatemodels,andmulti-delitysurrogates.InChapter 3 ,IexposeissuestowhetherornotuseMFsurrogates.InChapter 4 Idescribethephenomenonstudied,Iexplainthedetailsofoursimulations,Idescribethemetricsselectedtomeasurethedeparturefromaxisymmetry,andIpresentthemetricsdependenceontheperturbationvariablesconsidered.InChapter 5 Ishowhowsymmetries 20

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canbetakenintoaccountwhenbuildingsurrogatemodels.InChapter 6 Iexplorethedierenceandsimilaritiesbetweenthemetricsconsidered.InChapter 7 IconstructMFsurrogatestopredictthemetricsandcomparetheirperformancewithsingle-delitysurrogates.TheperformanceofMFsurrogateswithandwithoutconsideringparametricsymmetrieswasalsocompared. 21

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CHAPTER2LITERATUREREVIEW 2.1MutiphaseDetonationandNon-classicalInstabilitiesItisbelievedthatthegrowthofinitialperturbationsthatisobservedintheexplosivedispersalofparticlesisatleastinpartduetoRayleigh-Taylor(RT)[ 25 ]andRichtmyer-Meshkov(RM)[ 26 27 ]instabilities.TheRTinstabilityoccurswhenaninterfacebetweenaheavyandlightuidacceleratesinthedirectionoftheheavyuid,orequivalentlywhentheinterfacedeceleratesinthedirectionofthelighteruid.TheRMinstabilityariseswhenashockwaveorotherimpulseaccelerationtravelpastadensityinterface.Inbothcases,theinstabilitygrowthisintermsofvorticalstructuresthatappearduetothebaroclinicproductionofvorticity.Furthermore,intheidealsetupofaplanardensityinterfaceofinnitesimalthickness,theinitiallineargrowthisexponentialandthegrowthratescalesasthesquarerootofthewavenumberoftheperturbation.Thus,theshortwavelengthsaretheonestogrowmostrapidlyandthesizeofthemostampliedmodeislimitedbyviscousandotherdiusionalmechanisms.TheRTandRMinstabilitieshavethepotentialtorapidlygrowthanyinitialperturbationpresentintheparticlebed.However,thepresenceofnite-sizedinertialparticles,thesupersonicbackgroundowandthepresenceofothercompressibleowfeaturesmaycausethistoactasnon-classicalRTandRMinstabilities.ItisalsowellestablishedthatthegrowthrateoftheRTinstabilityshiftsfromanexponentialgrowninthelinearregimetoanalgebraicgrowthandsaturationinthenon-linearregimeoncetheperturbationhasgrownsuciently.Thistransitionfromlineartonon-linearbehaviortypicallyoccurswhentheamplitudebecomeslargerthanroughlyfortypercentoftheperturbationwavelength[ 28 ].TheRTinstabilityofsphericallyexpandinggasfrontsresultingfromasphericalshocktubeproblemhasbeenstudiedbothintheinviscidandviscousowregimesusinglinearstabilityanalysis[ 29 30 ].Theseresultshavealsobeenextendedtocylindricalsystems[ 31 ]whichalsoshowedthatdespitetheaddedcomplexityoftheowresultingfroma 22

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sphericalorcylindricalshocktube,thesimpletheoryofEpstein[ 32 ]isabletopredictboththeexponentialandalgebraicgrowthbehaviorremarkablywell.Similarly,theroleofRMinstabilityforsingleandmultimodalperturbationhasbeenstudiedingreaterdetailforagasonlysystem[ 33 ].Ofmorerelevancetothepresentstudyisthelinearinstabilityanalysisperformedtoinvestigatetheinstabilityofradiallyexpandingparticulatefronts[ 30 31 ].Largerinstabilitiesmaybeinuencedbythefragmentingofthecontainerwhichholdsthehigh-energyexplosive.However,instabilitiesalsoclearlyformintheabsenceofthesecasings[ 4 ].ToexploretheRMinstability,Ripleyetal.[ 4 ]performedanumericalstudywhichspeciedinitialperturbationsusingshallowellipticaldimplesintheedgeofacircularchargewithaspacingfrequencycorrespondingtothenumberofexperimentallyobservedinstabilities.Thisstudydemonstratedthattheinitialedgeperturbationandasimpleparticleinteractionmodelleadtotheformationofcoherentinstabilities.Xuetal.[ 34 ]showedusingmesoscalesimulationsthatthenumberofparticleinstabilitiesisdictatedbytheinitialparticlenumberintheinnerlayerattheexplosiveinterface.Inanattempttolinkthisndingtothemacroscopicrealworld,Ripleyetal.[ 4 ]interpretedtheinner-layerparticlesasparticlefragmentsofacasingbetweentheexplosiveandpackedparticlebed.Theinstabilityinducedbythefragmentsfromtheinnerboundarydictatedthelate-timenumberofmajorinstabilitystructures[ 35 ].Unsteadyeectsarealsoknowntoplayasignicantroleinmultiphaseowsundercertainowconditions.Whetherornottheyhaveanimpactonthegrowthofthemixinglayerisanopenquestion[ 36 ]andisaphenomenonofinterestinthisstudy. 2.2StrategiesforDesignofExperimentinMulti-delitySurrogateModelsHFmodelsusuallyrepresentthebehaviorofthesystemtoacceptableaccuracyfortheapplicationintended.Thesemodelsareusuallyexpensiveandtheirmultiplerealizationsoftencannotbeaorded.LFmodelsarecheaperandlessaccurate.Theyare 23

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obtained,forexample,bydimensionalityreduction,linearization,simplerphysicsmodels,coarserdomains,partiallyconvergedresults,etc.,seeFigure 2-1 Figure2-1. LFmodelsarecheaperbecausetheyareusuallyasimplicationofHFmodels. MFmodelscombineboth,LFmodelsandHFmodels,andhavedrawnmuchattentioninthelasttwodecadesbecausetheyholdthepromiseofachievingthedesiredaccuracyatlowercost.MFmodelsinvolve,generally,theconstructionofsurrogatemodels.Surrogatemodelsareapproximationscreatedtoreducecomputationalcostwhenalargenumberofexpensivesimulationsareneededforsuchprocessesasoptimization(e.g.[ 14 16 ])anduncertaintyquantication(UQ)(e.g.[ 24 ]).Surrogatemodelsconstructedusingdatafromdierentdelitiesarecalledmulti-delity(MF)surrogatemodels.Surrogatemodelscanalsobeconstructedtoreducethecostoftheindividualmodels.WhensurrogatemodelsarettedtotheHFdata,thenumberofsamplesneededforanaccurateapproximationmaystillrequireanunfordableamountofcomputation.Apossiblesolutiontothisproblemistorelyonlower-costLFsimulations.Theyareoftenthetypeofsimulationsthatwereusedtoanalyzesimilarproblemsagenerationagowhencomputersweremuchlesspowerful.Surrogatemodelscanalsobeconstructedto 24

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approximatetheLFmodels,butusuallythesearecheapenoughtousethemodeldirectly,seeforexampleNguyenetal.[ 37 ].ConstructinganMFsurrogatebycombiningdierentdelitylevelsisnotmandatoryforusingMFmodels,seeforexampleChoietal.[ 38 ],wheredierenttypesofdelityareusedecientlythroughadaptivesamplingandnoMFsurrogateisconstructed.ThesealternativeMFmodelsarecalledMFhierarchicalmodels.Figure 2-2 showsthetwopossibleoptionsfortheconstructionofanMFmodel.AnMFmodelwhereasurrogateisconstructedtocombinethedelitiesiscalledMFsurrogatemodel,otherwise,ifnosurrogateisconstructedandthedelitiesarecombinedinahierarchicalmanner,iscalledMFhierarchicalmodel.EachoftheseusesHFandLFmodelsortheirsurrogates. Figure2-2. IfMFmodelsinvolvetheconstructionofasurrogatemodeltoexplicitlycombinedelities(e.g.co-Kriging)itiscalledMFsurrogatemodel.Ontheotherhand,ifthedelitiesarecombinedinahierarchicalmannerwithoutbeingexplicitlycombinedinasurrogatemodel(e.g.importancesampling)itiscalledMFhierarchicalmodels. AlthoughthevastmajorityofthepapersreviewedinthissectionlimitedtheMFmodeltotwodelities,MFcanbeconstructedusingmorethantwodelities,forexample 25

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inHuangetal.[ 39 ],Forresteretal.[ 14 ],Qianetal.[ 40 ],LeGratiet[ 41 ],andGohetal.[ 42 ].Multi-levelmethods,wheretheHFmodelismerelyreplacedbyanLFmodel(withapossibleperiodiccheckonaccuracy),arenotconsideredasMFmodelinthisreview.ThesemodelreductionmethodsspeedupprocessessuchasoptimizationwithapayoofreducedaccuracywhileMFmodelsareabletoobtainanequilibriumbetweenthedesiredaccuracyandtheaordablecost[ 18 ].Althoughcostreductionwhilemaintainingthedesiredlevelofaccuracysoundsveryattractive,MFmodelsoftenrequireasubstantialinvestmentoftimeandeortonthepartoftheuseranditisnotclearfromtheliteraturewhenthepayojustiestheeort.Buildingsurrogatemodelsrequiresasamplingstrategyforthegenerationofarepresentativegroupofsamplepoints.Samplingstrategiesarealsorelatedtotheaccuracythatthesurrogatemodelwillachieve,seeDribuschetal.[ 43 ].Thesimplestsamplingmethodsaregrid-based,suchasfullfactorialdesign(FFD)whereeachvariable(factor)issampledataxednumberoflevels.Thismethodisusedforlowdimensionalproblems(usuallylessthanthreevariables),seeFigure 2-3A .ItsapplicationcanbeseeninFernandez-Godinoetal.[ 44 ].Centralcompositedesign(CCD)methodtakesthetwo-levelFFDandaddstoittheminimumnumberofpointsneededtoprovidethreelevelsofeachvariablesothataquadraticpolynomialcanbetted.Itisoftenusedwhenthenumberofdesignvariablesisbetweenthreeandsix,seeFigure 2-3B .ForhigherdimensionproblemsonlyasubsetoftheverticesoftheCCDisusedinanapproachcalledsmallcompositedesign[ 45 ].FFD,CCD,andSCDarenotexibleinthenumberofsamplingpointsanddomainshape.Designsofexperimentsthatallowanynumberofsamplesareusuallybasedonanoptimalitycriterion.Forexample,inD-optimaldesign[ 46 ]asubsetofagridinanydomainshapeisselectedbymaximizingthedeterminantoftheFisherinformationmatrix[ 47 ].Thisreducestheeectofnoiseonthettedpolynomialleadingtomost 26

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A BFigure2-3. FFDandCCDsamplingstrategies.A)Fullfactorialdesign(FFD)with3factorsand4levels;B)Centralcompositedesign(CCD)with3factorsand5levels. ofthepointsbeingattheboundaryofthedomain.Figure 2-4 showstheapplicationofD-optimalcriterioninanestedsamplingdesignformulti-delity(MF)surrogates.Space-llingmethodsthatspreadthepointsmoreuniformlyinthedomainaremorepopularwhenthenoiseinthedataisnotanissue.Whenthereissubstantialnoise,thebestmethodistosamplenearthedomainboundariesusinganoptimalitycriterionmethod.Space-llingmethodsincludeMonteCarloandLatinhypercubesampling(LHS).ThemostcommonavorofLHSattemptstomaximizetheminimumdistancebetweenpoints,alsoknownasmaximincriterion[ 48 ],inordertopromoteuniformity.WhenitcomestoMFsurrogates,thereistheadditionalissueoftherelationbetweenthelow-delity(LF)andhigh-delity(HF)samplingpoints.NesteddesignsamplingstrategygeneratesHFpointsasasubsetofLFpointsorLFpointsasasupersetofHFpoints.Itwasinitiallydevelopedasaspace-llingmethodforgeneratingadditionaldatasetstocomplementtheexistingoneusingacriterion.ForexampleJinetal.[ 49 ]use 27

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threeoptimalitycriterion,maximindistancecriterion,entropycriterionandcenteredL2discrepancycriterion.TheunionoftheoriginalsamplingpointsandtheadditionalonesbecomesthesamplingpointsforasurrogatemodelbuiltusingLFsamples,whiletheadditionalsubsetisusedfortheconstructionofasurrogatemodelusingHFsamples[ 50 ].HaalandandQuian[ 51 ]proposenesteddesignsamplingforcategoricalandmixedfactors.Zhengetal.[ 52 ]comparednestedandnon-nesteddesignsamplingtoexploretheirrespectiveeectsonmodelingaccuracy.HavingtheHFdatapointsasasubsetoftheLFdatapointsmakestheparameterestimationeasierformethodsthatbuildadiscrepancyfunction.AdiscrepancyfunctionisanadditivecorrectionconstructedusingtherelationshipbetweenLFandHFdatatoestimatetheHFresponse.Iftheyarenotasubset,theparameterestimationofthediscrepancyfunctionbecomesdependentontheparameterdeterminationoftheLFsurrogatemodel.Forinstance,co-KrigingmethodmodelsuncertaintiesusingGaussianprocessforboththeLFsurrogatemodelandthediscrepancyfunction.IfthedesignofexperimentssatisesthenestedsamplingconditionparametersofeachGaussianprocessmodelcanbeestimatedseparately.Nevertheless,thisisnotvalidforeveryMFsurrogateand,forexample,samplingpointsforBayesianinferencecannotsatisfythenestedcondition.However,ifweonlyconsidertheuseofMFsurrogatesforcombiningcomputersimulationresults,wecancontroltheinputsettingsofsimulationsandthereforesatisfythenestedcondition.Thearemultiplenesteddesignschoices,onepossibilityistorstgeneratethedesignofexperimentsfortheLFsurrogateandthenselectasubsetusingsomecriterion.ThismethodisusedinBalabanovetal.[ 53 ]wheretheygenerated2107pointsin29-dimensionalspaceusingSCDfortheLFsamplingpointsandthenselected101HFsamplingpointsusingtheD-optimalitycriterion.Itisalsopossibletotaketheopposite 28

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methodandgenerateLFpointsasasupersetoftheHFpoints.AnexampleofD-optimalcriterionisshowninFigure 2-4 Figure2-4. Nestedsamplingdesign.LFdatapoints(bluebubbles)areplacedrstandthen,usingD-optimaldesign,theHFdatapoints(orangebubbles)areselected. LeGratiet[ 54 ]generatedindependentlytheLFandHFsamplingpointsandthentheLFnearestpointtoeachHFpointismovedontopoftheircorrespondingnearestneighbor,asillustratedinFigure 2-5 .Thismethodisusuallycallednearestneighborsampling. Figure2-5. Nearestneighborsampling.HFpoints(bluebubbles)andLFpoints(orangebubbles)aresampledindependently,thentheLFnearestneighborpointtoeachHFpointismovedontopofit(blackbubbles). Adaptivesamplingmethodsarestrategiesusedtoreducethenumberofsimulationsrequiredtoconstructamodelwithaspeciedaccuracyusingeectiveinterpolationandsamplingmethods.Thesemethodsarewidelyappliednowadaysanddierentoptionscanbefoundintheliterature.Inparticular,Mackmanetal.[ 55 ]comparedtwoadaptivesamplingstrategiesforgeneratingKrigingandradialbasissurrogatemodels.Theyfoundthatbothperformbetterthantraditionalspacellingmethods. 29

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IthasrecentlybecomepopulartouseLFsurrogatesandreducedordermethodsinlocalsearchesofparameterspaceforoptimalplacementofnewdesignpointsaswecanseeinRobinsonetal.[ 56 ],andinRaissiandSeshaiyer[ 57 ]. 2.3SurrogateModelsSurrogatemodelsareapproximationsthatarettotheavailabledataandmakeafunctionalrelationshipbetweeninputvariablesandtheoutputquantityofinterest.Surrogatemodelsarewidelyusedwhileconstructingmulti-delitymodels.Sometimestheyareconstructedforeachdelityseparatelyinamulti-delityhierarchicalmodelmethod,forexampleinNelsonetal.[ 58 ],andKozielandLeifsson[ 59 ].HeretheMFmodelistheecientmannerthatthesesurrogatemodelsareusedinordertoimproveaccuracy.Alternatively,theinformationofdierenttypesofdelitiescanbeincludedinasinglesurrogate,forexampleinGiuntaetal.[ 60 ],Qianetal.[ 40 ],andPadronetal.[ 1 ].Mostsurrogatemodelsarealgebraicmodelsthatapproximatetheresponseofasystembasedonttingalimitedsetofcomputationallyexpensivesimulationsinordertopredictaquantityofinterest.Theaccuracyofasurrogatemodelisalsodeterminedbythedesignofexperimentusedtoselectthedatapoints,thesizeofthedomainofinterest,thesimulationaccuracyatthedatapointsandthenumberofsamplesavailable[ 19 ].Peherstorferetal.[ 18 ]includeacompletesectionofprojection-basedmodelsanddata-tmodelswherethereadercanextendtheinformationincludedinthissection.Responsesurfacemodels(RSM)areoneoftheoldestsurrogatemodelsandtheymaystillbethemostwidelyusedinengineeringdesign.RSMarettedbylinearregressioncombiningsimplicityandlowcostasitonlyrequiresthesolutionofasetoflinearalgebraicequations.RSMusuallyassumesthatthefunctionalbehavior(e.g.asecondorderpolynomial)iscorrectbutthedatapointsresponsehasnoise.InMFcontext,RSMcanbefoundinalargenumberofpapers,justtocitesomeofthem,Changetal.[ 61 ],Burgeeetal.[ 62 ],Venkatarmanetal.[ 63 ],Balabanovetal.[ 53 ],Balabanovetal.[ 64 ],Masonetal.[ 65 ],Vitalietal.[ 66 ],Knilletal.[ 67 ],Vitalietal.[ 68 ],Umakantetal.[ 69 ], 30

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Venkatarmanetal.[ 70 ],Choietal.[ 38 ],Sharmaetal.[ 71 ],Sharmaetal.[ 72 ],Sunetal.[ 73 ],Goldsmithetal.[ 74 ]andChenetal.[ 75 ].Polynomialchaosexpansion(PCE)becamepopularinthiscenturyfortheanalysisofaleatoryuncertaintiesusingprobabilisticmethodsinuncertaintyquantication(UQ)[ 76 { 78 ].InPCE,thestatisticsoftheoutputsisapproximatedbyconstructingapolynomialfunctionthatmapstheuncertaininputstotheoutputsofinterest.Thechaoscoecientsareestimatedbyprojectingthesystemontoasetofbasisfunctions(Hermite,Legendre,Jacobi,etc.).InMFcontext,PCEapplicationscanbefound,forexample,inEldred[ 79 ],NgandEldred[ 24 ],Padronetal.[ 80 ],Padronetal.[ 1 ],andAbsiandMahadevan[ 81 ].Withincreasingcomputerpower,moreexpensivesurrogatemodelsbecamepopular.TheseincludeKriging,articialneuralnetworks(ANN),movingleastsquares(MLS)andsupportvectorregression(SVR).Oneoftheadvantagesoftheseisthattheyusuallyworkbetterforhighlynon-linear,multi-modalfunctions.Krigingsurrogatemodelestimatesthevalueofafunctionasthesumofatrendfunction(e.g.polynomial)representinglow-frequencyvariation,andasystematicdeparturerepresentinghigh-frequencyvariationcomponents[ 82 ].UnlikeRSM,mostKrigingapproachesassumethatthedatapointresponseiscorrectbutthefunctionalbehaviorisuncertain.Kriginghasbecomeaverypopularsurrogate,ingeneral,butevenmoresoinMFapplications.Thismayreectthefactthatithasanuncertaintystructurethatlendsitselftonon-deterministicMF.ApplicationsofKrigingsurrogatesintheMFcontextcanbefoundinLearyetal.[ 83 ],Forresteretal.[ 14 ],Gohetal.[ 42 ],Biehleretal.,Huangetal.[ 84 ],Biehleretal.[ 85 ],andFidkowskietal.[ 86 ].Co-Kriging[ 87 88 ]iscommonlyknownastheextensionofKrigingtoincludemultiplelevelsofdelitiesinthesurrogateconstruction.Applicationsoftheco-KrigingmethodcanbefoundinChungandAlonso[ 89 ],Forresteretal.[ 90 ],Yamazakietal.[ 91 ]andHanetal.[ 92 ].LaurenceandSagaut[ 93 ]comparedKrigingandco-Krigingperformance. 31

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ANNsconsistofarticialneuronsthatcomputeaweightedsumofinputsandasaturationfunctionandthencomputestheoutputofthearticialneuron.AnexampleofANNapplicationinMFcanbefoundinMinisciandVasile[ 21 ]whereitisusedduringtheoptimizationprocesstocorrecttheaerodynamicforcesinthesimpliedLFusingacomputationaluiddynamicsHFmodel.TheLFisusedtogeneratesamplesgloballyovertherangeofthedesignparameters,whiletheHFisusedtolocallyrenetheANNsurrogatemodelinlaterstagesoftheoptimization.Anotherwell-knownsurrogatemodelisMLSsurrogate,whichwasintroducedbyLancasterandSalkauskas[ 94 ]andwasextensivelydiscussedinLevin[ 95 ].MLSisanimprovementoftheweightedleast-squaresmethod(WLS)proposedbyAitken[ 96 ].WLSrecognizesthatalldesignpointsmaynotbeequallyimportantinestimatingthepolynomialcoecients.AWLSmodelisstillastraightforwardpolynomial,butwiththetbiasedtowardspointswithahigherweighting.InanMLSmodel,theweightingsarevarieddependinguponthedistancebetweenthepointtobepredictedandeachobserveddatapoint.ExamplesofitsimplementationinMFcanbeseeninToropovetal.[ 97 ],Zadehetal.[ 98 ],Zadehetal.[ 99 ],Bercietal.[ 100 ],andSunetal.[ 101 ].Traditionalsurrogatemodelspredictscalarresponses.Somenontraditionalonessuchasproperorthogonaldecomposition(POD)areusedtoobtaintheentiresolutioneldtoapartialdierentialequation(PDE).Toal[ 102 ],Rodericketal.[ 103 ]andMifsudetal.[ 104 ]exploreMFmodelPODmethodinFluidMechanics. 2.4SomeStatisticsaboutMulti-delityModelsThroughoutthiswork,wereviewedalargevarietyofMFimplementations,andwehavechosenaclassicationsystembasedonsixattributes,seeFigure 2-6 .Thecategoriesaretheapplication,thedelitytype,themethodusedtoconstructtheMFsurrogatemodel(deterministicmethod(DM)andnon-deterministicmethod(NDM)),theyearpublished,thepapereld,andthesurrogatemodelused.Figure 2-6 givesthereadera 32

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senseofhowthesecategoriesaredistributedthroughouttheliteraturereviewed.Theattributesaredescribedasfollows: ApplicationreferstothekindofproblemsolvedusingMFmodels.Threemainapplicationswerefound,optimization,uncertaintyquantication,andoptimizationunderuncertainty.Nonereferstothepapersthatdescribeagenericprocedurewithoutanyapplication. Typesofdelityreferstothenatureofthedelitywherephysicsreferstoadierenceinassumptionsandconsiderationsinthephysicalmodel(e.g.Euler-BernoullibeamtheoryasLFmodelvs.TimoshenkobeamtheoryasHFmodel),numericalsolutionaccuracyreferstodierentlevelsofdiscretizationinspaceortimeandalsotopartiallyconvergedsolutions;numericalmodelsreferstowhenthesamephysicalmodelandassumptionsareusedbutsomethinginthewaythattheresultsarecomputedchanges(e.g.2DReynolds-AveragedNavier-Stokes(RANS)simulationsasLFMvs.3DRANSsimulationsasHFmodel);andSim+Expreferstothecombinationofsimulations,usuallyasLFmodel,andexperiments,usuallyasHFmodel,intheconstructionofanMFmodel. MethodreferstothecriterionusedtotthedataintheMFsurrogatemodelconstruction(DMorNDM).NonereferstopapersthatuseMFhierarchicalmodelswherenoMFsurrogatemodelisconstructed. Yearpublishedreferstotheyearwhenthepaperwasreleased. Fieldreferstotheareaoftheproblemsolvedinthepaper.ThemostcommoneldsfoundwereFluidMechanicsandSolidMechanics. SurrogatemodelreferstothesurrogatemodelusedtoconstructtheMFsurrogatemodel.NonerepresentsthepapersthatuseMFhierarchicalmodelswithoutconstructinganMFsurrogatemodel.Figure 2-6A showsthatthemostcommonapplicationfoundforMFmodelsisoptimization,followedbyUQandbyoptimizationunderuncertainty.Theseapplicationsareintroducedasouter-loopapplicationsbyPeherstorferetal.[ 18 ]andextensivelydiscussedinSection5,6and7oftheirwork.ThefactthatoptimizationisthemainapplicationisunderstandablebecauseUQandoptimizationunderuncertaintyarerelativelynewsubjects.However,itisexpectedthatmorepublicationswillappearintheseapplicationsinthenearfuture. 33

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A B C D E FFigure2-6. ProportionofdierentattributesconsideredintheMFmodelpapersreviewed,thechartsarebasedon178papers.A)Application;B)TypesofFidelity;C)Method;D)YearPublished;E)Field;F)SurrogateModel. 34

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Figure 2-6B showsthedistributionofpapersbythetypeofdelitiesused;thesearediscussedinSection 2.4.1 .Themostcommondierenceindelitiesfoundintheliteratureisduetophysicsfollowedbynumericalsolutionaccuracy,e.g.griddiscretization.Figure 2-6C showsthattheproportionofpapersthatusedeterministicmethods(DM)andnon-deterministicmethods(NDM)fortheconstructionofanMFsurrogatemodelissimilar.ThecategoryNonereferstopapersthatpresentMFmodelswithoutconstructinganMFsurrogatemodel,i.e.MFhierarchicalmethods.Thisisthecase,forexample,inoptimizationwhereLFmodelsareusedtoreducethedomainofinterestandthenHFmodelsareusedtodeterminemoreaccuratelywheretheextremeis,seeRodriguezetal.[ 105 ]andPeherstorferetal.[ 106 ].ThemostcommonmethodsusedtocombinedelitiesinMFcontextarepresentedinSection 2.4.2 .Figure 2-6D showsthattheuseofMFmodelsseemstobeexpandingsinceitsbeginninginthelate90's.InSection 2.4.3 afurtherstudyofthetimedistributionofDMandNDMispresented.MFmodelscanbeusedtoreducethecostforagivenaccuracyorimproveaccuracyforagivencomputationcost.Figure 2-6E showsthatmostofthepapersreviewedapplyMFmodelsintheeldsofFluidMechanicsandSolidMechanics.OtherincludesElectronics,Aeroelasticity,andThermodynamics.Nonerepresentspaperswithoutanyspecicapplication(e.g.somepapersusedmathematicalfunctionslikeHartmanorRosenbrocktotestthemethods).Figure 2-6F showsthedistributionofpapersbysurrogatetype.ThetwomainlyusedsurrogatesforMFsurrogateconstructionwerefoundtoberesponsesurfaceandKriging.ThecategoryOthersincludesarticialneuralnetworks(4%),movingleastsquares(2%),properorthogonaldecomposition(POD)(2%),andsupportvectormachines,radialbasisinterpolationandwithlessthan1%each.MFmodelswithouttheconstructionofanMFsurrogatemodel,MFhierarchicalmodels,areincludedinthecategoryNone.E.g.Choietal.[ 38 ]proposedhierarchicalMFmodelsforoptimizationwhereHFmodelsareonlyusedwhentheyareneededtocorrecttheshortcomingsoftheLFmodel.HerenoMF 35

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surrogatemodelsarebuiltanddelitiesarenotexplicitlycombined.AnotherexampleisKalivarapuandWiner[ 107 ]whereanMFmodelisusedforinteractivemodelingofadvectiveanddiusivecontaminanttransportwithnoMFsurrogatemodelconstruction.OtherexamplesareGiuntaetal.[ 60 ]andZahiretal.[ 108 ]. 2.4.1TypesofFidelityIntheliteraturereviewed,wefoundthatthedierenttypesofdelitiesarecommonlyassociatedwithfourprincipalcategories: Physics:Simplifyingthemathematicalmodelofthephysicalreality,typicallychangingthedierentialequationsbeingsolved.Forexample,modelingaowusingEulerinviscidequationscorrespondstoalowerdelitymodelandmodelingtheowusingRANS(Reynolds-averagedNavier-Stokes)equationscorrespondstoahigherdelitymodelandbyintroducingturbulenteects.Alternatively,thelowerdelitycanrepresentasimplicationofthenumericalmodel.Examplesincludelinearizationbysimplifyingthegeometrysothatthedimensionalityoftheproblemcanbereduced,andsimplifyingtheboundaryconditionstoallowasimplersolution. NumericalSolutionAccuracy:Changingthediscretizationmodel,suchasusinglowergriddiscretizationorpartiallyconvergedresultsastheLFmodel. NumericalModels:Samephysicalmodelandassumptionsareusedbutsomethinginthewaythattheresultsarecomputedchanges(e.g.2DReynolds-AveragedNavier-Stokes(RANS)simulationsasLFmodelvs.3DRANSsimulationsasHFmodel). SimulationandExperiments:Usingexperimentalresults.Inthiscase,experimentsareconsideredthehighestdelity.Generally,wecanclearlystatewhichdelityishigher(e.g.nevs.coarsergridwhileusingthesamemodel),butsometimesthisisnotanoption(e.g.1Dmodelwithnegridvs.the3Dmodelwithcoarsergrid).Figure 2-7 summarizestheinformationabove.MFmodelscanbeusedinmanydisciplinesandthedelitiesinvolvedcanvarydependingontheapplication.TheMFmodelsfoundinthepapersreviewedaregenericand,althoughtheyweredevelopedinacertainarea,canbeusedinmultipleelds. 36

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Figure2-7. Maindierencesbetweendelitiesfoundintheliterature. However,intheliteraturereviewwehavefoundtwomaineldswhereMFmodelsareused,FluidMechanicsandSolidMechanics.InFluidMechanicsthemainmodelsfoundwerebasedonanalyticalexpressions,empiricalrelations,numericallinearapproximations,potentialow,numericalnon-linearnon-viscousapproximations(Euler),numericalnon-linearviscousapproximations(RANS),coarsevs.renedanalysisandsimulationsvs.experiments.Table 2-1 showspapersthatusethesemodelsastheLFmodelandtheHFmodel.Table 2-2 includesextracategoriesfoundinFluidMechanics:dimensionality(e.g.2D/3D),coarsevs.renedanalysis,simulationsvs.experiments,transientvs.steadyandsemi-convergedvs.convergedsolutions.OthermodelsthatarenotincludedintheTable 2-1 orTable 2-2 are: SimplifyingphysicsfoundinCastroetal.[ 152 ],whereanearthpenetratorproblemissimpliedbyassumingarigidpenetrator. InGoldfeldetal.[ 153 ],wherethephysicsaresimpliedbyassumingconstantinsteadofvariablematerialproperties. InForresteretal.[ 148 ]wheretheLFmodelisaRANSsimulationwithsimpliedgeometryandtheHFmodelisaRANSsimulationwithfullgeometry. InKeane[ 154 ],wherethedelitydistinctionisbasedonthenumberofMonteCarlosamplestobecombined. 37

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Table2-1. FluidMechanicsorientedpapersperLFmodelandHFmodelused(An:analytical,Em:empirical,Li:linear,PF:potentialow,Eu:Euler,RANS:Reynolds-averagedNavier-Stokes) ReferenceAnEmLiPFEuRANS [ 74 ][ 69 ]LF-HF---[ 109 ][ 110 ][ 111 ][ 112 ][ 23 ][ 14 ]-LFHF--[ 21 ][ 113 ][ 37 ][ 114 ]-LF---HF[ 62 ][ 115 ][ 116 ][ 67 ][ 117 ][ 118 ][ 119 ][ 80 ][ 120 ]--LF-HF-[ 121 ][ 122 ][ 123 ][ 52 ][ 124 ]LFHF[ 125 ][ 126 ][ 58 ][ 127 ]---LF-HF[ 128 ][ 129 ][ 130 ][ 84 ][ 1 ][ 131 ]---LFHF InSolidMechanicsthemainmodelsfoundwereanalyticalexpressions,empiricalrelations,numericallinearapproximations,numericalnon-linearapproximations,andcoarsevs.renedanalysis.Table 2-3 showspapersthatusethesemodelsastheLFmodelandtheHFmodel.AnothermodelnotincludedinthetableisfoundinKimetal.[ 155 ],whereLFmodelandHFmodelareisothermalandnon-isothermalanalysis,respectively.Table 2-4 includesadditionalmodelsfoundinSolidMechanicsincludingdimensionality(e.g.2D/3D),coarsevs.rened,simulationsvs.experiments,andboundaryconditionsimplication(e.g.inniteplatevs.niteplate).SomepaperswhoseeldwasnotFluidorSolidMechanicswerealsoreviewed;thesepapersarelistedbelow: 38

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Table2-2. FluidMechanicsorientedpapersbyLFmodelandHFmodelused.Thecategoriesaredimensionality(e.g.2D/3D),coarsevs.renedanalysis,simulationsvs.experiments,transientvs.steadyandsemi-convergedvs.convergedsolutions.ThephysicalmodelusedbyeachpaperwasalsoassignedwhereEm:empirical,Li:linear,PF:potentialow,Eu:Euler,RANS:Reynolds-averagedNavier-Stokes,URANS:unsteadyRANS,TM:turbulencemethod,MHD:magnetohydrodynamics,AE:aeroelasticequations,MFF:multiphaseowandTM:thermomechanicalequations FidelityTypeReference Dimensionality[ 23 ]2D/3DEu,[ 132 ]1D/3DRANS+TM,[ 126 ]2D/3DURANS,[ 133 ]2D/3D,[ 40 ]1D/2DRANS,[ 134 ]1D/2DLi,[ 135 ]1D/3DRANS,[ 136 ]1D/3DRANS,[ 20 ]1D,2D/3DRANSCoarse/Rened[ 137 ]Eu,[ 138 ]RANS,[ 139 ]Eu,[ 121 ]Eu,[ 38 ]Li/Eu,[ 22 ]RANS,[ 140 ]MFF,[ 141 ]MHD,[ 142 ]Eu,[ 59 ],Eu[ 143 ]Eu,[ 144 ]Eu,[ 145 ]RANS,[ 146 ]RANS,[ 147 ]RANS,[ 108 ]Eu/RANSExp./Sim.[ 86 ]Euler/MHD,[ 148 ]PF/Em,[ 149 ]RANS,[ 150 ]RANSSemiconverged/Converged[ 22 ],RANS[ 59 ]EuSteady/Transient[ 100 ]AE,[ 151 ]Eu,[ 147 ]TM, Table2-3. SolidMechanicsorientedpaperspertypeofanalysisusedtodeterminedelity(An:analytical,Em:empirical,Li:linear,NL:non-linear) ReferenceAnEmLiNL [ 71 ]LF-HF-[ 70 ][ 63 ]-LFHF-[ 63 ]-LF-HF[ 156 ][ 157 ][ 158 ][ 159 ][ 120 ][ 105 ][ 160 ][ 161 ]--LFHF 39

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Table2-4. SolidMechanicsorientedpaperspertypeofdelityusedbesidesanalysistype.Thecategoriesaredimensionality(e.g.2D/3D),coarsevs.renedandboundaryconditionsimplication(e.g.inniteplatevs.niteplate).ThemodelusedbyeachpaperwasalsoassignedusedwhereLi:linear,NL:non-linear FidelityTypeReference Dimensionality[ 162 ]1D/2DLi,[ 163 ]1D/3D,[ 164 ]2D/3D,[ 65 ]2D/3DLi,[ 72 ]2D/3DLi,[ 165 ]2D/3DNLCoarse/Rened[ 53 ]Li,[ 85 ]NL,[ 166 ]Li,[ 167 ]NL,[ 61 ]Li,[ 83 ]Li,[ 168 ]Li,[ 73 ]NL,[ 101 ]NL,[ 169 ]Li,[ 98 ]Li,[ 99 ]LiBoundaryConditions[ 66 ]Li,[ 170 ]Li InElectronicsthemostcommonmethodiscoarsevs.renedanalysis(Koziel[ 171 ],KozielandOgurtsov[ 172 ],Jacobsetal.[ 173 ])althoughAbsiandMahadevan[ 81 ]usedsteadyvs.transientmodels. InRobotics,Winneretal.[ 174 ],thedelitiescorrespondedtocomplexitydeterminedbyresourcesavailabletotherobot. Someofthepaperstesttheirmethodsusingmathematicalfunctionsandthereisnotanapplicationtoaparticulareld.Forexampleanalyticalfunctionvs.analyticalapproximationsofthefunctionareshowninRobinsonetal.[ 56 ][ 175 ],ZimmermannandHan[ 176 ],Ngetal.[ 24 ],LeGratiet[ 54 ],RaissiandSeshaiyer[ 177 ],RaissiandSeshaiyer[ 57 ]andGohetal.[ 42 ]. Inthecategoryofmethodsforuncertaintyanalyseswithnoapplicationtoaeldinparticular,wefoundBurtonandHajela[ 178 ],Eldred[ 79 ],Perdikarisetal.[ 9 ],Peherstorferetal.[ 106 ],andChaudhuriandWillcox[ 179 ].InBurtonandHajela,inEldred,andinPerdikarisetal.thetypesofdelitywerelessandmoreaccurateuncertaintyanalysis.InPeherstorferetal.[ 106 ]LFmodelswereusedtoaidintheconstructionofthebiasingdistributionforimportancesamplingandasmallnumberofHFsamplesareusedtogetanunbiasedestimate.ChaudhuriandWillcox[ 179 ]employedaniterativemethodthatusedLFsurrogatemodelsforapproximatingcouplingvariablesandadaptivesamplingoftheHFsystemtorenethesurrogatesinordertomaintainasimilarlevelofaccuracyasuncertaintypropagationusingthecoupledHFmultidisciplinarysystem. 40

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2.4.2MethodsforCombiningFidelities 2.4.2.1Multi-delitysurrogatemodelsvs.multi-delityhierarchicalmodelsInthissurvey,MFsurrogatemodelsrepresent59%(122/173)ofthepapersreviewed(Figure 2-8 ).In41%(51/173)ofthecasesMFsurrogatemodelsarenotconstructed,andinstead,thedierenttypesofdelitiesarecombinedusingsomecriteriontobenetaprocess,suchasoptimization,inapproachesthatwehavecalledMFhierarchicalmodels.Forexample,BurtonandHajela[ 178 ],Choietal.[ 139 ],andSinghandGrandhi[ 165 ]useHFsurrogatemodelsonlywhenneededbecauseLFmodelshaveexhaustedtheirrangeofcapability.AnotherexampleisChristenandFox[ 180 ]wheretheLFmodelisusedforMarkovChainMonteCarlosamplingandtheHFmodelisonlyusedwhentheacceptancecriterionismet.TheacceptancecriterionisbasedonthelikelihoodfunctionconstructedusingtheLFmodel.SimilarworkwasdoneinEbyetal.[ 181 ],Drissaouietal.[ 158 ]andNarayanetal.[ 182 ].InRethoreetal.[ 146 ]thelargestpartoftheoptimizationisperformedusingsimpler/fastercostfunctionsandcoarseresolutionandincreasingtheresolutionofthedomainandthecomplexityofthemodelswhereneeded.Narayanetal.[ 182 ]usedastochasticcollocationprocedurewheretheLFmodelsareevaluatedextensivelytoselectthedatapointstobeevaluatedthroughtheHFmodel.Peherstorferetal.[ 106 ]useimportancesamplingmethodbasedonanLFmodeltochoosethesamplingpointsfortheconstructionoftheHFsurrogatemodel.Peherstorferetal.[ 18 ]categorizethemethodstocombinedelitiesinadaptation,fusion,andltering.AdaptationenhancestheLFmodelwithinformationfromtheHFmodelwhilethecomputationproceedsandtheSMareadaptedineachiteration.MethodsbasedonfusionevaluateLFmodelandHFmodelandthencombineinformationfromalloutputs,anexampleoffusionisco-Krigingmethod[ 90 ].FilteringmethodsinvoketheHFmodelfollowingtheevaluationofanLFmodellter.Thatis,theHFmodelisusedonlyiftheLFmodelisinaccurate,orwhenthecandidatepointmeetssomecriterionbasedontheLFmodelevaluation.Peherstorferetal.[ 18 ]includeMFhierarchicalmodelsinthe 41

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Figure2-8. Ofthetotalofthe178papersreviewed,127constructedamulti-delitysurrogatemodeltoexplicitlycombinethedelities.TherestofthepaperspresentanMFmodelusingmulti-delityhierarchicalmodels. categoriesadaptationandlteringmanagementmethods.Thereadercanrefertotheirworkforfurtherinformation.FusionmethodsareincludedinourMFsurrogatemodelscategory. 2.4.2.2Multi-delitysurrogatemodelsOurattentionwasfocusedontheMFmodelthatconstructasurrogatemodeltoexplicitlycombinedelitieswhicharecalledMFsurrogatemodels.MFsurrogatemodelsmainconceptistouseanalgebraicsurrogatetocorrecttheLFmodelusingtheHFmodel.Fourmaincorrectionmethodsare:multiplicativecorrection,additivecorrection,comprehensivecorrection,andspacemapping.Insomecases,theparametersoftheLFmodelaredierentfromthoseoftheHFmodel.ThereforeatransformationisneededfromLFparameterstoHFparameters.ExamplesaregiveninRobinsonetal.[ 134 ]andKozieletal.[ 183 ].Inconventionalmathematicalmethodsifrst-orderconsistency(i.e.theLFmodelanditsderivativematchtheHFmodel)issatisedbetweentheLFmodelandtheHFmodel(e.g.Alexandrovetal.[ 128 ],andAlexandrovetal.[ 137 ])wecanassureconvergence.Ifsecond-orderconsistencyissatisedconvergenceratescanbeimproved 42

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(Eldredetal.[ 184 ]).Ontheotherhand,meta-heuristicoptimization(notmathematicallyrigorous),whichingeneralhasslowerconvergenceandaccuracy,ispreferredinglobaloptimization(KavehandTalatahari[ 185 ]). Additiveandmultiplicativecorrections:OnepossibilityistocorrecttheLFmodelresponsebyconstructingasurrogatemodelofthedierenceortheratiobetweentheHFmodelandtheLFmodel,calledadditiveormultiplicativecorrectionsrespectively.TheestimatoroftheHFmodel,theMFsurrogatemodel,usinganadditivecorrectiontocorrecttheLFmodel,canbeexpressedas ^yHF=yLF(x)+(x):(2-1)where(x)isasurrogatemodelcalledadditivecorrection,alsoknownasdiscrepancyfunction,whichisbasedonthedierencebetweentheHFmodelandtheLFmodel.TheestimatoroftheHFmodel,theMFsurrogatemodel,usingamultiplicativecorrectioncanbeexpressedas ^yHF=(x)yLF(x)(2-2)where(x)isthemultiplicativecorrection,whichisasurrogatemodelconstructedusingtheratiobetweenHFmodelandLFmodel.Alexandovetal.[ 137 ]constructedanMFsurrogatemodelusingmultiplicativecorrectionsinaerodynamicoptimizationproblems.Balabanovetal.[ 53 ]comparedtheperformanceofanMFsurrogatemodelconstructedusingadditiveandmultiplicativecorrectionsforasimilaroptimizationproblem.Forresteretal.[ 23 ]correctedpartiallyconvergedresultsusinganadditivecorrectionbasedonfullyconvergedresults.ThereadercanndmorereferencestoMFsurrogatemodelconstructedusingadditiveandmultiplicativecorrectionsinTable 2-5 andTable 2-6 inSection 2.4.3 .Figure 2-9 isaschematicexampleofmultiplicativeandadditivecorrectionsforthecasewhere(x)and(x)areconstant.AfterthecorrectiontheestimateoftheHFmodelhasimproved.TheratioyHF/yLForthedierenceyHF-yLFatthesamplingpointsasfunctionsofthedesignvariablevectorxareusedtoobtainthemultiplicativeoradditivecorrectionsrespectively.IftheLFmodelisnotcheapenough,yLF(x)canbealsoreplacedbyanLFsurrogatemodel.Thereisnosinglewaytoobtainacorrectionfactorandthefollowingexampleillustratestwopossibleoptions.Supposethatwecanaordonly20HFmodelrealizationsand200LFmodelrealizations.TherststepistobuildasurrogatetoapproximatethedierenceortheratiobetweentheLFmodelandHFmodelanalysesbasedonthe20nesteddatapoints.Forthesecondstep,wehavetwooptions: 43

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A BFigure2-9. Schematicofconstantcorrectionfactors. 1. Buildasurrogateusingthe200LFdatapoints,thentheMFsurrogatemodelwouldbethesumoftwosurrogates(ifweusedthedierence)ortheproduct(ifweusedtheratio). 2. Usethesurrogatebuiltintherststeptoapproximatethediscrepancy,ortheratio,atthe180datapointswhereonlyLFdatapointsaregiven.ThencalculatethepredictedHFmodelresultsatthese180datapointsasthesumofthediscrepancy,orratio,calculatedpreviouslyandtheLFmodelestimation.WenowhaveHFmodeldatapointsat20samplingpointsandwehaveestimatedHFmodeldataat180points.Wetreatthemequallyandtasurrogatemodeltothe200pointsusingthissurrogateasanMFsurrogatemodel.Thedierencebetweenthesetwooptionscanbenoticeablewhenweuseregressionratherthaninterpolation.Withtherstoption,wecangetlargedierencesbetweentheMFmodelpredictionsandtheHFmodeldataatpointswhereHFmodeldataisgiven.Withthesecondoption,thedierencemaybesmaller,andwecanalsomakeitevensmallerbyusingaweightedleastsquare(WLS)surrogatewithhigherweightsforHFmodeldata. Comprehensivecorrections:Acomprehensivecorrectionisalsopossible,wherebothcorrections(additiveandmultiplicative)areusedinthesameMFsurrogatemodel.Themultiplicativecorrectionisinmostcasesaconstant,seeforexampleKeane[ 154 ]andPerdikarisetal.[ 9 ].Apossiblecomprehensivecorrectioncanbewrittenas, ^yHF=(x)yLF(x)+(x)(2-3)whereisthemultiplicativecorrectionsurrogate,andistheadditivecorrectionsurrogate.Ourliteraturereviewshowsthatthemostcommonmethodistosetthemultiplicativefactorasaconstantandtouseasurrogatemodeltoapproximate 44

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theadditivecorrection,howeverwefoundacomprehensivecorrectionwithnon-constant,thatis(x),inQianetal.[ 40 ].AnothercomprehensivecorrectionfoundintheliteratureisthehybridmethoddevelopedbyGanoetal.[ 186 ], ^yHF=w(x)(x)yLF(x)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(w(x))[yLF(x)+(x)](2-4)wherew(x)isaweightingfunction.ThismethodisusedforexampleinZhengetal.[ 187 ]andFischeretal.[ 188 ].InTable 2-5 andTable 2-6 ofSection 2.4.3 ,thereadercanndfurtherreferenceswhereMFsurrogatemodelsarebuiltusingcomprehensivecorrections.Finally,athirdcomprehensivecorrectionthatwecanconsiderisspacemapping.InsteadofcorrectingtheoutputoftheLFmodel,itisalsopossibletocorrecttheinputvariablesinamethodcalledspacemapping.SpacemappingwasrstintroducedbyBandleretal.[ 189 ][ 190 ]andthekeyideabehindthismethodisthegenerationofanappropriatetransformationofthevectorofnemodelparameters,xHF,tothevectorofcoarsemodelparameters,xLF, xLF=F(xHF):(2-5)Thistechniqueallowsthevectors,xHFandxLF,tohavedierentdimensions.FindingthisrelationshipFisaniterativeprocess,anditisdesirable,althoughnotnecessary,forFtobeinvertible.ThegoalisthattheHFmodelresponse,yHF(xHF),andtheLFmodelresponse,yLF(xLF)satisfy kyHF(xHF))]TJ /F8 11.955 Tf 11.95 0 Td[(yLF(xLF)k(2-6)withinsomelocalregion,wherekkisasuitablenormandisatolerancesetting.Thecombinationofdelitiesusingspacemappingwasonlyfoundindeterministicmethods(DM).Therstreviewpaperofspacemappingmethodwaspublishedaftertenyearsofthespacemappingimplementation[ 191 ]andthesecondoneaftertwodecades[ 192 ].Thespacemappingconcepthasbeenextendedtoincludeaggressivespacemapping[ 193 ],trustregions[ 194 ],articialneuralnetworks[ 195 ],implicitspacemapping[ 196 ],neural-basedspacemapping[ 197 ][ 198 ],inverseproblems[ 199 ],correctedspacemapping[ 134 ]andtuningspacemapping[ 183 ].Table 2-5 ofSection 2.4.3 providesfurtherliteraturewherespacemappingisusedtoconstructMFSMs. 2.4.3DeterministicMethods(DM)vs.Non-deterministicMethods(NDM)WhileDMassumebasisfunctionsandndtheircoecientsbyminimizingthediscrepancybetweenthedataandthefunction(e.g.Vitalietal.[ 68 ],Goeletal.[ 200 ]),NDMassumeanuncertainfunctionwhichparametersaredeterminedmaximizingthelikelihoodfunction.(LeGratietandCannamela[ 201 ]),seeFigure 2-10 .DMcanbe 45

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appliedtoanysurrogatebecausetheydonotneedanuncertaintystructureasNDMdo;ontheotherhand,NDMwerefoundtobemoreaccuratethanDMinKeane[ 154 ]andinPark[ 50 ]. Figure2-10. MFsurrogatemodelsparameterscanbedeterminedusingdeterministicmethods(DM)ornon-deterministicmethods(NDM)dependingontheassumptionsandtheprocedureforthedeterminationoftheunknownparameters. Whenwearedealingwithouter-loopapplications,suchasuncertaintyquantication,NDMrequireamethodofstatisticalinferencetotreatparameteruncertaintiestoavoidtheexpensivestandardMonteCarlosimulations[ 202 ][ 203 ].ThemostpopulariscalledBayesianframeworkwheretheposteriordistributionoverthemodelparametersdependsonthelikelihoodandthepriordistributionoftheunknownparametersviaBayesrule.Gaussianprocessisaexible,convenientandwidelyusedclassofdistributionstomodelpriorknowledgeaboutourdata[ 204 ].AnalternativetotheGaussianprocesscanbefoundinKoutsourelakis[ 205 ]wheretheuncertaintiesaremodeledusingnon-Gaussiandistributions.FortheDM,thescalarisestimatedtominimizethedierencebetweenthepredictionoftheLFsurrogatemodel(^yLF)andtheHFsurrogatemodel(^yHF)atthehigh-delitydatapoints.Meanwhile,NDM,suchasBayesiandiscrepancyorco-Kriging,estimateascalarthatmakesthediscrepancyfunctionassimpleaspossibleevenifitincreasesits 46

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magnitude.Bysimplifying,theaccuracyofthediscrepancysurrogatecanbehigherthanbyminimizingthediscrepancy.Intheengineeringcommunity,calibrationhasbeenwidelyusedtoimprovesimulationpredictionsbyadjustingphysicalparameterstoachievethebestagreementwithexperiments(KosonenandShemeikka[ 206 ],Owenetal.[ 207 ],Leeetal.[ 208 ],McFarlandetal.[ 209 ],Coppeetal.[ 210 ],YooandChoi[ 211 ]).AlthoughwedonotconsiderpurecalibrationasanMFmodel,KennedyandOHagan[ 204 ]presentapopularBayesiancalibrationmethodthathasadierentperspectiveoncalibration.CalibratedphysicalparametersusingBayesiancalibrationcanbedierentfromtheirtruevaluessinceittreatscalibrationparametersthesamewayasothernon-physicalhyper-parameters.IcallthismethodcalibrationalongwithcomprehensivecorrectionandweconsidereditanMFmodel.SomeofitsapplicationsarefoundinQianetal.[ 40 ]andBiehleretal.[ 85 ].Figure 2-11 showsthat55%ofthepapersthatconstructMFsurrogatemodelsusedeterministicmethods(DM)while45%usenon-deterministicmethods(NDM).ThegurealsoshowstheproportionofeachofthemethodsforMFsurrogateconstructionlistedinSection 2.4.2.2 .IconcludethatmultiplicativemethodsarethemostusedinDM,however,forNDMthemostcommonarecomprehensivecorrections(i.e.bothmultiplicativeandadditivecombined).The20th-centuryliteraturewasmostlydominatedbyDM.Inthe21stcentury,NDMusingKriging[ 212 ],co-Krigingmodels[ 168 ](Krigingextensiontomultiple-delitysetsofinputs)andrelatedsurrogatemodels[ 204 ]becamewellknowninthestatisticalcommunity.Figure 2-12 presentsahistogramthatshowsthepercentageofpaperspublishedperyearintervalforbothDMandNDM.Thepercentageiscalculatedwithrespecttothetotal(122papers)thatusesDMorNDMtocombinedelities.TheuncertaintypredictioninKrigingsurrogatescanbeconstructedusingtechniquessuchasgeneralizedleastsquaresorGaussianprocess(GP).AGPisacollectionofrandomvariableswiththepropertythatthejointdistributionofanynitesubsetis 47

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A B CFigure2-11. Ofthe127reviewedpapersthatconstructaMFSmodel,54%usedeterministicmethods(DM)while46%usenon-deterministicmethods(NDM).ThegurealsoshowsthedistributionofthetechniquesusedtocombinedelitiesinasingleMFSmodelforeachofthem.A)Method;B)DistributionofthetechniquesusedtocombinedelitiesinasingleMFSmodelfornon-deterministicmethods(NDM)foundinthereviewedliterature.C)DistributionofthetechniquesusedtocombinedelitiesinasingleMFSmodelfordeterministicmethods(DM)foundinthereviewedliterature Gaussian.Krigingsurrogates,constructedusingGP,becamealsoquitepopularduringthiscenturyaswecanseeforexampleinKennedyandO`Hagan[ 140 ]andinLeGratiet2012[ 54 162 163 ].ManypopularMFsurrogatemodels(e.g.co-Kriging,Bayesian-basedcomprehensivecorrection,etc.)useGaussianprocess(GP)tomodeleachdelityresponseanditspredictionuncertainty.ItisimportanttonotethatanMFsurrogatemodelusuallyisbasedonassumptionsand,iftheproblemdoesnotsatisfythem,theaccuracymaysuer. 48

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Figure2-12. Percentageofpapersreviewedpublishedfromearly90suntilthepresentfordeterministicmethods(DM)andnon-deterministicmethods(NDM).Thepercentageofpapersiscalculatedwithrespecttothetotal(127papers)reviewedthatusesDMorNDMtocombinedelitiesinanMFS.NDMpercentagehasbeenincreasingremarkablyinthelastyears. Forexample,themostcommonassumptionforGPbasedmethodsisthatthepredictionuncertaintyofonedelityisindependentofthepredictionuncertaintyintheotherdelity.Table 2-5 andTable 2-6 organizethepapersthatuseDMandNDM,respectively,fortheconstructionofMFsurrogatemodel. 49

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Table2-5. Papersthatusedeterministicmethods(DM)fortheconstructionoftheMFsurrogatemodel. CombiningMethodReference Additivecorrection[ 53 ][ 62 ][ 79 ][ 153 ][ 67 ][ 80 ][ 56 ][ 175 ][ 213 ][ 71 ][ 72 ][ 101 ][ 73 ][ 214 ][ 160 ][ 170 ]Multiplicativecorrection[ 215 ][ 137 ][ 53 ][ 111 ][ 62 ][ 61 ][ 153 ][ 216 ][ 217 ][ 159 ][ 218 ][ 145 ][ 65 ][ 182 ][ 56 ][ 175 ][ 71 ][ 72 ][ 101 ][ 73 ][ 214 ][ 70 ][ 63 ][ 66 ][ 161 ][ 170 ][ 169 ]Comprehensivecorrection[ 219 ][ 129 ][ 155 ][ 24 ][ 103 ][ 98 ][ 99 ]Spacemapping[ 152 ][ 22 ][ 143 ][ 131 ][ 134 ] Table2-6. Papersthatusenon-deterministicmethods(NDM)toconstructtheMFsurrogatemodel. CombiningMethodReference Additivecorrection[ 85 ][ 23 ][ 130 ][ 141 ][ 119 ][ 168 ][ 106 ][ 40 ][ 120 ]Multiplicativecorrection[ 75 ][ 148 ][ 118 ]Comprehensivecorrection[ 109 ][ 167 ][ 138 ][ 14 ][ 42 ][ 84 ][ 154 ][ 140 ][ 149 ][ 54 ][ 41 ][ 162 ][ 163 ][ 220 ][ 9 ][ 102 ][ 150 ][ 221 ]Calibration+comprehensivecorrection[ 222 ][ 86 ][ 223 ][ 204 ] 50

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CHAPTER3ISSUESINDECIDINGWHETHERUSINGMULTI-FIDELITYSURROGATES 3.1SummarySimulationsareoftencomputationallyexpensiveandtheneedformultipleevaluations,asinuncertaintyquanticationoroptimization,makessurrogatemodelsanattractiveoption.Forexpensivehigh-delitymodels(HFM),however,evenperformingthenumberofsimulationsneededforttingasurrogatemaybetooexpensive.Multi-delitysurrogates(MFS)areapopularwayofcombiningasmallnumberofhigh-delity(HF)samplesfromHFmodels(HFM)withalargenumberoflessaccuratebutcheaplow-delity(LF)samplesfromLFmodels(LFM)intoasinglesurrogate.MFSbecamefamousbecausetheypromisetoachievethedesiredaccuracyatasubstantiallyreducedcost.Ouraimistoseeifthereareanyindicationsunderwhatcircumstancesthissubstantialcostreductionisrealized.Inthiswork,wehavereviewedmorethan170papersandidentiedsomechallengesinMFSthatseemtohavenotbeenovercomeyet.Fromtheliterature,itappearsthatitishardtogetanidea,intermsofcostsavings,whenitisusefultoinvesttheadditionaleortofcreatingandusingMFS.EvenwhentheeortisinvestedandonehasbothLFandHFsamples,wedonotappeartohaveadependablewaytotellwhetherweshouldusetheLFsamplesbecausesometimesusingLFsamplescandomoreharmthangood.TheseandotherissuesrelatedtoMFSarepresentedanddiscussed. 3.2Nomenclature (x):discrepancyfunction,alsoknownasadditivecorrection (x):multiplicativefactor,alsoknownasmultiplicativecorrection :constantscalingfactor yH(x):high-delitymodel yL(x):low-delitymodel 51

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3.3BackgroundHigh-delitymodels(HFM)usuallyrepresentthebehaviorofasystemtoacceptableaccuracyfortheapplicationintended.Thesemodelsareusuallyexpensiveandtheirmultiplerealizationsoftencannotbeaorded.Ontheotherhand,low-delitymodels(LFM)arecheaperbutlessaccurate.Theyareobtainedbydimensionalityreduction[ 20 ],simplerphysicsmodels[ 21 ],coarserdiscretization[ 22 ],partiallyconvergedresults[ 23 ],etc.Multi-delitymodels(MFM)combinetheinformationofboth,LFMandHFM,andhavedrawnmuchattentioninthelasttwodecadesbecausetheyholdthepromiseofachievingthedesiredaccuracyatlowercost.Surrogatesareoftenusedforapproximationscreatedtoreducecomputationalcostwhenalargenumberofexpensivesimulationsareneededforsuchprocessesasoptimization(e.g.Forresteretal,2007[ 14 ]orVianaetal.,2014[ 16 ])anduncertaintyquantication(UQ)(e.g.NgandEldred,2012[ 24 ]).Surrogatescanbeconstructedtoreducethecostofsingledelitymodels.Surrogatemodelsconstructedusinginformationfrommodelsofdierentdelitiesareusuallyknownasmulti-delitysurrogates(MFS).Inthispaper,weconsiderMFSasasubcategoryofMFMbecauseMFScanbeconsideredasawayofcombiningtwodelitymodels.WhileMFMincludeanypossiblemannerofcombiningHFMandLFMtoreducecostatthedesiredaccuracy,MFScombineHFMandLFMthroughasurrogatemodel(e.g.,co-Kriging[ 14 ]).WecallMFhierarchicalmodelstothoseMFMthatarenotMFS.ThemostcommonMFhierarchicalmodelsuseLFMandHFMindependentlyguidedbyacriterion(e.g.Lazzaraet.al,2009[ 133 ],Jolyet.al.,2014[ 132 ]andDurantinetal.,2017[ 224 ]).ThemainobjectiveofthisreviewpaperistoinvestigatetheusefulnessofMFMandMFSintheperspectiveofcostsavingsandaccuracyimprovement.MFSoftenrequireaninvestmentoftimeandeortforimplementation.However,itisunclearfromtheliteraturewhenthepayojustiestheeort.ThereistheadditionalquestionofwhetherMFSshouldbeconstructedevenifnoeortisrequired.AsZhangetal.,2017a[ 225 ]and 52

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Guoetal.,2018[ 226 ]haveshown,attimesusingtheLFsamplesleadstoasurrogatethatislessaccuratethanoneusingonlyHFsamples.Thepaperisorganizedasfollows.InSection 3.4 weshowthatsubstantialprogresshasbeenachievedinthepastfewyearsintheaccuracyofMFSbycombiningadditiveandmultiplicativecorrectionfactors.InSection 3.5 wediscussandcompare,inthecontextofoptimizationandbasedon19papers,thecostratiobetweenLFMandHFManalysesandthecostratiooftheentireoptimizationprocessusingMFMandHFM.EarlierMFStypicallyusedonlyoneortheother.InSection 3.6 wesummarizeMFMndingsinsomerecentpapers,showingthefollowingchallenges: HowcanwedecidewhentouseonlytheHFsamplestoconstructasurrogateratherthanfusingtogethertheHFandtheLFsamplesinanMFS? HowcanwechoosebetweenmultipleLFM? HowtousebothadditiveandmultiplicativefactorsinMFMapproximationforsurrogatesotherthanco-Kriging?BasedontheextensivereviewofpapersemployingMFS,itwasfoundthatusuallyliteraturereportsthesuccessofusingMFSbutrarelydiscussesthecauseofthesuccess.ThisisinspiteofsubstantialprogressintheaccuracyofMFS.InSection 3.7 wesuggestaguidelineforreportingcomputationalcost-eectivenessbyusingMFM.Sincecostsavingisofgreatinteresttomanydierentelds,reportingitwouldhelpotherresearchersindecidingwhetherornottouseMFMintheirapplications.Inaddition,weincludedthreeappendices.Appendix 2.2 discussesdierenttechniquesfordesignofexperimentinMFScontext.Appendix 2.3 introducessurrogatemodelstothereader.Atlast,Appendix 2.4 includesawidecompilationofMFMstatisticsbasedontheliterature. 3.4ProgressinFittingSurrogatestotheDataInthissection,wereviewtherecentprogressonMFS.ThereadercanrefertoAppendix 2.3 fordetailedinformationaboutsingledelitysurrogatemodels.MFSare 53

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surrogatesmodelsbuiltusingmorethanonedelity.EarlyMFSmostlyusedadditiveormultiplicativecorrections(e.g.Alexandovetal.,2001[ 137 ],Balabanovetal.1998[ 53 ],Forresteretal.,2006[ 23 ]).Thatis,givenalow-delityfunctionyL(x)andahigh-delityfunctionyH(x)therelationbetweenthetwowasassumedas yH(x)=yL(x)+(x);(3-1)or yH(x)=(x)yL(x):(3-2)Thatis,therelationshipbetweenLFandHFwasconsideredadditiveormultiplicative.OnlythedierenceortheratiobetweenLFandHFareusedtotthediscrepancyormultiplicativecorrection,respectively.OftentheLFM,yL(x),issocheapthatanLFsurrogateisnotevenneeded.However,ifasurrogateisneeded,theLFsurrogatemodelcanbeeasilybuiltwithalargenumberofLFsamplesbecausethecostofobtainingthemislow.FittingthesurrogateforthemultiplicativeoradditivecorrectionismorechallengingbecauseitsaccuracydependsonthesmallnumberofHFsamples.Toillustrateoneofthedicultiesassociatedwiththeseapproaches,wewillusethefollowingexample.LettheHFMyHbe yH(x)=4x2+2;0x1(3-3)andtheLFMyL(x)be yL(x)=3x2+4;0x1:(3-4)Figure 3-1 showsboth,yHandyL,asafunctionofx.NotethattheHFfunctionvariesintherange[2,6],andtheLFfunctionintherangeof[4,7].Forthesefunctions,theexactLFcorrectionsare (x)=x2)]TJ /F1 11.955 Tf 11.96 0 Td[(2;0x1;(3-5) 54

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Figure3-1. Low-delity,yL(x)),andhigh-delity,yL(x),functions. ifweusetheadditivecorrection,or (x)=4x2+2 3x2+4;0x1;(3-6)ifweusethemultiplicativecorrection.Theadditivecorrectionisaquadraticpolynomialvaryingintherange[-2,-1],whilethemultiplicativecorrectionisarationalfunctionvaryingintherangeof[0.5,0.86].Therangeofthecorrectionsisimportantbecause,withasmallnumberofHFsamples,wecanexpectthatcorrectionerrorsareasignicantfractionoftherangeofthecorrection.Forexample,iftheerrorisaboutone-thirdoftherange,thenfortheadditivecorrectionthatwouldmeananerrorintheorderof1=3,whichwouldtranslatetoanerrorofabout17%atx=0.Forthemultiplicativecorrection,wecanexpectanerroroftheorderof0.12,soatx=0wemayget0.62correctioninsteadof0.5,oranerrorof24%.However,thepredictionerrorisnotjustafunctionofthecorrectionrange.Itcouldbealsoafunctionoftheshape(i.e.,simplicity)ofdiscrepancy,andthecorrelationbetweenLFMandHFM.Considertwofunctionsy=sin(x)andy=xinx=[0;10].Therangeof 55

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therstfunctionismuchsmallerthanthatofthesecondbutthesecondismucheasiertot.AsubstantialimprovementinaccuracywasachievedbytheintroductionofascalarmultipliertotheLFfunction.ThefollowingformofMFSiscalledthecompressiveapproach: yH=yL(x)+(x):(3-7)Forourexample,=4=3willleadtoaconstantexactcorrection=)]TJ /F1 11.955 Tf 9.3 0 Td[(10=3.Withaconstantcorrection,onewouldexpectanexactresult.Ofcourse,therightvalueofneedstobeestimated.However,evenanapproximatevalue,say=1:25,wouldreducethecorrectionto=0:25x2)]TJ /F1 11.955 Tf 12.49 0 Td[(3,reducingtherangeof,andhencetheexpectederrorsbyafactorof4.Unfortunately,mostMFSusersdonottakeadvantageofthescalarfactor.Ofthe127reviewedpapersthatconstructedanMFS,only30usethecomprehensiveapproach.Thatis,boththeadditiveandmultiplicativecorrectionsareusedinthesamesurrogate.SeeFigure 2-11 ,Appendix 2.4 .Parketal.(2017[ 50 ],2018[ 227 ])lookedatthesuccessofthecombinationofGaussianProcessorKrigingsurrogateswithBayesianidenticationofandbyusingthemaximumlikelihoodestimation.TheyconcludedthattheBayesianapproachtendstominimizethebumpinessofsothatitcanbettedaccuratelywithasmallnumberofavailableHFsamples.Inotherwords,weimprovethecorrelationbetweenHFMandLFMbyminimizingthebumpiness.Inonedimensionthebumpinessbofafunctionf(x)isdenedastheintegralofthesquareofthesecondderivativeas b=Zjf00(x)j2dx:(3-8)Inhigherdimensions,Parketal.2018[ 227 ]averagedthebumpinessoveralargenumberoflinesinrandomdirections.Fortheexamplesthattheyexamined,theminimumerrorwasverycloseinvaluetotheonethatminimizesbumpiness. 56

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3.5CostRatiovs.SavingsAsarststage,weexaminedtheecacyofMFMforcostandtimesavingswhilemaintainingthedesiredaccuracy.SincecostandtimesavingsarethemaingoalofusingMFM,itwouldbeappropriateforresearcherstoreportintheirpublicationsthecostsavingsbyusingMFM.Unfortunately,thisdoesnotseemtobeacommonpracticeinthecommunity.Despitethisdiculty,wewereabletocollect,fromsomepublicationsthatusedMFMforoptimization,thecostratiobetweenperformingasingleanalysisofLFandHF,andthecostratiobetweenperformingoptimizationusingMFMandusingHFM.Inotherwords,thereportedcostintimeassociatedwiththeentireoptimizationprocessifMFMareuseddividedbythesamecostbutusingHFMiscalledMF/HFoptimizationcostratio.ThereportedcostintimeforasingleanalysisoftheLFMdividedbythesamecostbutfortheHFMiscalledLF/HFanalysiscostratio.InFig. 3-2 wepresenttheMF/HFoptimizationcostratioasafunctionoftheLF/HFanalysiscostratio.TheinformationpresentedinFig. 3-2 wasextractedfrom20papersoutofthe102reviewedthatperformoptimizationinwhichboththeMF/HFoptimizationcostratioandthecostandLF/HFanalysiscostratiocanbecalculated.OnewouldexpectthatcomputationalsavingswouldbeenhancedwhentheLFcostsareasmallfractionoftheHF.However,thegureshowsthatthereisnotaclearrelationshipbetweenLF/HFanalysiscostratioandMF/HFoptimizationcostratio.Becausewecouldnotndaclearrelationship,wespeculatethattherelationbetweencostandaccuracyoftheLFMinvolvedmightplayabigrole.Thatis,veryinexpensivemodelstendtobelessaccurateandtheoptimizationconvergencecanbedelayedduetothisfact.Meanwhile,amoreexpensiveLFMcanmaketheoptimizerconvergefastermakingthisabetteroption.Also,havingahighcorrelationbetweentheHFMandLFMmightleadtobetterMFSapproximation.AhighcorrelationbetweenHFMandLFMwillusuallyresultinahighlyaccurateMFMbecausethediscrepancybetweenthemcanbeeasilypredicted, 57

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thereforepredictingtheHFMbecomeseasier.Inaddition,thecomplexityoftheresultingmodelmayalsoinuencethecostoftheoptimization. Figure3-2. CostratiobetweenasingleanalysisoftheLFMandasingleanalysisoftheHFMvs.costratiobetweentheoptimizationprocessusinganMFMandtheoptimizationprocessusinganHFM.ThedottedlineseparatestheregionwhereperformingtheoptimizationusingMFMresultsincostsavings(upperoctant)andtheregionwheretherearenotcostsavings(loweroctant). InordertounderstandtherelationshipbetweenLF/HFanalysiscostratioandMF/HFoptimizationcostratiobetter,Fig. 3-3 showsthesamedataasinFig. 3-2 buthighlightingeld,approachtocombinethedelities,approachtoobtainsingleormulti-delitysurrogatescoecients,andtypeofsurrogateused.Overall,theredoesnotseemtobeanystraightforwardrelationshipbetweenthedata.Figure 3-3A marksupthedierentapplicationeldswheretheoptimizationwasperformed.OptimizationmainlyfocusedonuidmechanicsseemstobethemostcommonamongtheonesthatuseMFMbutwealsohavefoundsolidsmechanicsandelectronics. 58

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A B C DFigure3-3. Eectofthesurrogatesused.Thereisnoclearrelationshipbetweenthecasesthatusethesamesurrogate.A)Eectoftheeld.Thereisnoevidentrelationshipifwecompareonlycasesappliedtothesameeld;B)Eectoftheapproachused.Thecorrelationbetweendeterministicapproachesdoesnotseemstrong.Samehappenswithnon-deterministicapproaches;C)Eectofthecombinationmethodused.Noobvioustrendsassociatedtocasesthatusethesameapproachtocombinedelitiesisobserved;D)ResponseSurface,Kriging,Co-Kriging,SupportVectorRegression,PolynomialChaos,NoSurrogates. 59

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Fromthegure,itdoesnotappearthattheapplicationelddeterminestheusefulnessofMFMforoptimization.Figure 3-3B distinguishesbetweendeterministicandnon-deterministicapproaches.Deterministicapproachesminimizethedierencebetweenthedataandthet.Innon-deterministicapproaches,thelikelihoodthatthedataisconsistentwiththetismaximized.Thereisnoobviousrelationshipbetweenthegainsobtainedusingdeterministicapproaches,non-deterministicapproachesorboth.Figure 3-3C identiespapersbytheapproachusedtocombinedelities.AdditiveormultiplicativeapproachesrefertotheonesthatpredicttheHFMcorrectingtheLFMusingadiscrepancyfunctionoramultiplicativefactor,respectively.Comprehensiveapproachesaretheonesthatcombineadditiveandmultiplicativecorrections.HierarchicalmodelsdonotexplicitlybuildamodelcombiningLFandHFbutusebothindependently.Thesemodelsuseanalgorithmorcriteriontodecidewhentousethem.Spacemappingcorrectstheinputvariablesinspace,insteadoftheoutput,topredicttheHFM.Noobvioustrendsassociatedwiththecasesthatusethesameapproachtocombinedelitiesisobserved.Fig. 3-3D highlightsindierentcolorsthesurrogatemodelsused,showingthatthereisnoclearrelationshipbetweenthecasesthatusethesamesurrogate.Responsesurfacemodels(RSM)exploretherelationshipsbetweenindependentvariablesandoneormoreresponsesusingadesignofexperiments.RSMwereinitiallydevelopedtomodelexperimentalresponses(BoxandDarper,1987[ 228 ])andlatertomodelnumericalresponses(vanCampenetal.,1990[ 229 ],Toropovetal.,1996[ 230 ],Giuntaetal.1997[ 231 ]).Kriging(Matheron,1963[ 232 ]basedonKrige'swork[ 233 ])isaninterpolationmethodwherethedatapointsareinterpolatedbyaGaussianprocessmodelobtainedbymaximizingthelikelihoodfunctionofthemodelforthegivendata.Co-KrigingisthegeneralizationofKrigingformultiplesetsofdata.SupportVectorRegression,developedbyVapniketal.mainlyatAT&Tlaboratoriesinthenineties[ 234 235 ],allows 60

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tointroduceerrorboundsalongwiththedataanditndsapredictionthathasanassociatederrorestimation.Ifinterested,thereadercanrefertotheoverviewofthesesurrogateapplicationsinoptimizationinForester,2007[ 14 ].Polynomialchaosisawayofrepresentinganarbitraryrandomvariableofinterestasafunctionofanotherrandomvariablewithagivendistribution,andofrepresentingthatfunctionasapolynomialexpansion.ItwasrstintroducedbyWeinmerin1938[ 236 ]andgeneralizedbyXiuin2010[ 237 ].Polynomialchaosexpansionisoftenusedinoptimizationunderuncertainty.SincetheliteraturedoesnotshowaclearrelationshipbetweenLF/HFanalysiscostratioandMF/HFoptimizationcostratio,wecannotmakeaconclusionwhentouseMFMtosavecost.TheMFMsavings,however,canbehighlyproblemdependent.Unlesswearedealingwithaclassofproblemsofsimilarstructure,thesavingsthatanauthorreportsforoneproblemcouldbeverydierentfromthatforotherproblems,evenifthesamemethodologyisused.Thisissueismoreseverewhenthesavingsarenotjustduetothesurrogateconstructionbutforanentireoptimizationprocess.Forinstance,somealgorithmscanguaranteeconvergence,meaningthatanalgorithmwillconvergetoalocalcriticalpointofanHFproblemregardlessoftheinitialguess.However,therateofconvergencewilldependontherelativepropertiesoftheLFMandHFM. 3.6ChallengesinFittingMulti-delitySurrogates 3.6.1DecidingwhethertoUsetheLow-delityDataItisnotagiventhatMFSwillalwaysleadtobetteraccuracythanasurrogateusingonlytheHFdata.AsrecentlyreportedbyZhangetal.,2017a[ 225 ],inastrengthpredictionproblem,unexpectedlythesurrogatebuiltbyonly3ormoreHFsampleswasfoundtohavebetteraccuracythananMFSwiththesame3samples,aidedby12LFsamplesinatwo-dimensionalproblem.ThismeansthatgivenHFandLFsamples,weneedacriteriontodecidewhethertousetheLFsampleswithanMFSortotasurrogatetothegivenHFdataonly. 61

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UsingtheadditivecorrectionwithinEq.( 3-7 ),itappearsasifthemaximum-likelihood(ML)estimationcouldselect=0,thatwouldcorrespondtodisregardingtheLFdata.However,Guoetal.,2018[ 226 ]testedtheMLestimationforatwo-design-variableturbineproblemwithasingleHFMandtwoalternativeLFM.Here,twounsteadyReynolds-averagedNavier-Stokes(RANS)equationsolverswereused;afulltransientmodel(HF),atransientrotorblademodelwithtimetransformation(LF1),andasteadyRANSsolver(LF2).Themaindierencebetweensteady-stateandtransientmodelswastheirsettingsoftheinterfacebetweenstatorandrotor.Thedierencebetweenthetwotransientmodelswastheturbulencemodel.TheshearstresstransportationcoupledbytransitionisnotavailableintheLF1model;thereforeitsaccuracywouldbepoorerthantheHFmodel.TheyvariedthenumberofHFsamplesrangingfrom4to12with20dierentdesignofexperimentsforeachnumberofevensamples.Therefore,atotalof100designofexperimentswereconducted.Foreachcase36LF1sampleswereused.TheMFSwasmoreaccuratethantheHFsurrogatemodelfor59ofthe100forthebetterLFM(LF1),andfor18outof100forthepoorerLFM(LF2).However,MLcorrectlyselected=0foronly3ofthe123cases(41caseswhenusingLF1and82whenusingLF2)wheretheHFMwasmoreaccurate.Guoetal.alsotestedcross-validation(CV)forthesamepurpose.CVwasabletoidentify67ofthese123cases.Ananalysisofthefailuresdiscussedinthepreviousparagraphindicatedthat,asinthecaseoftheexampleofthequadraticfunctionofEq.( 3-3 )andEq.( 3-4 ),thebumpinessofthecorrectionwassubstantiallylowerthanthatoftheHFfunction,becauseofitslowerrangeofvariation.ThispredisposedboththeMLandCVestimationinfavoroftheMFS.Therefore,MLandCVestimationsarenotaccuratetodeterminewhetherLFdataareusefulornot. 1Thelowdelitysampleswereavailableata66grid,whichprecludedusingthemorecommonMFSdesignsofexperimentsdescribedinAppendix 2.2 62

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Thus,itappearsthatwemaystilllackadependablecriteriontotellwhetherwegainbyusingtheLFdata.Ofcourse,therearemanycaseswheretheMFSshouldbeclearlymoreaccurate,asforexample,mostofthecaseswhenwehaveonlyasingleHFsample.However,moreresearchintothechoicebetweenHF-alonesurrogateandMFSisclearlycalledfor. 3.6.2ChoosingbetweenMultipleLow-delityDatasetsWhentherearemultipleLFmodels/datasetsavailable,severaloptionsarepossible:usingallLFmodels,usingonlyasubsetofLFmodels,usingthebestLFmodel,ornotusinganyLFmodels.ThischallengeisrelatedtothepreviouschallengeofdecidingwhethertousetheLFdata.ThestudybyGuoetal.(2018[ 226 ])indicatedapossibleprobleminchoosingbetweentwoLFdatasourcesforanMFS.Intheirstudy,LF1wasmoreusefulthanLF2becauseithadabettercorrelationwiththeHFdatasothatthediscrepancyfunctionhadasignicantlylowerrangeofvariation(Section 3.4 ).Outof100designsofexperiments,theMFSbasedonLF2surrogateweremoreaccuratethantheMFSbasedonLF285times.Inaddition,theMLestimationcouldnotidentifyasinglecaseofthe15exceptionalcases,andtheCVestimationonlysixoutofthe15.ThisstudyindicatedthattheperformanceofMFSdependsonthedesignofexperimentsinadditiontothequalityofLFmodels.Moreover,MLandCVestimationsarenotgoodatchoosingabetterLFmodelforagivendesignofexperiments.Thisoneexampleisnotnecessarilyaproofthatthedicultyiscommon.However,itisanindicationthatthechoicebetweenmultipleLFMmaybeachallengedeservingofmoreresearch.AnotheroptionistouseallavailableLFMinasingleMFSasinChaudhurietal.,2018[ 238 ]andPeherstorferetal.,2018[ 239 ]). 3.6.3SelectingforOtherSurrogatesInSection 3.4 ,itwasshownthatintroducingthescalingfactorsignicantlyimprovedtheperformanceofMFS,byreducingthebumpinessinadditivecorrection.Theconstantscalingfactorcanbefoundbyminimizingthebumpinessoftheadditive 63

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correctionorbyminimizingtheerrorbetweenthescaledLFpredictionswiththatofHFsamples.Parketal.(2017[ 50 ])showedthattheformerperformsbetterthanthelatterbecauseitisecienttotasimpleadditivecorrectionwithasmallnumberofHFsamples.TheyalsoobservedthatmaximizingthelikelihoodfunctionintheGaussian-Processsurrogatesissimilartoreducingthebumpinessintheadditivecorrection.However,theMFSusingtheGaussian-Processsurrogatesrequiresuncertaintymodelforprediction;notallsurrogatesprovideone.WhileKrigingorGaussian-Processsurrogatesareoftenthemostaccurateoratleastclosetothemostaccurate,therearecaseswhenothersurrogatesaremoreaccurate(e.g.Vianaetal.,2009[ 240 ]).Especially,Krigingdoesnotworkwellwithaverynoisyresponseorwithaverylargenumberofsamples,especiallywhensomeofthemaretightlyclustered.Whennon-KrigingsurrogatesareusedforLFSandadditivecorrection,itisunclearhowthebumpinesscanbereducedeectively.OnemayuseaKrigingmodelforobtainingbutthenswitchtoadierentsurrogateforttingtheMFS.Thisapproachmaydeservefurtherstudy.Inaddition,itispossibletotreattheLFmodelasabasiswiththescalingfactorasanunknowncoecientinlinearregression.ThiswassuggestedbyZhangetal.(2017b[ 241 ]),butthereisnotmuchexperiencereportedonhowwellitworks. 3.7RecommendationsTimesavingswhilemaintainingthedesiredaccuracyisenoughofanincentiveforapplyingMFM.Unfortunately,wefoundthatitisoftendiculttotellfromapaperhowusefultheMFMimplementationwastoaccomplishthisgoal.Amongtheliteraturereviewedtobuildthispaperwefoundonepaper,Padronetal.,2016[ 1 ],thatweconsideragoodexampleofanexhaustivesavingsreport.Here,cost,savings,andaccuracywerestated.Reportingthecost,savings,andaccuracyoftheresultingMFMwillallowfutureuserstodecidewhetherornottouseMFMtobuildapproximationsoftheirownproblem. 64

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Table 3-1 presentsthesavingsreportextractedfromPadronetal.,2016whereanairfoilshapewasoptimizedusingsequentialquadraticprogramming.Theirgoalwastominimizetheaverageandstandarddeviationoftheairfoildragcoecientwhilemaintainingthedesiredliftcoecient.RANSCFD(Computationaluiddynamics)wasusedasHFMandEulerCFDasLFM.TheRANSHFMhada23,315pointsmeshwhere256wereontheairfoil.Ontheotherhand,theEulerLFMhada6,983pointsmeshwhere128wereontheairfoil.Thesurrogatemodelusedwasastochasticpolynomialchaosexpansionandthemodelswerecombinedthroughanadditivecorrection.ThecostofasingleLFanalysisis15timeslowerthanthatofasingleHFanalysis.TheerrorintheLFMwas18%oftheHFM.TwoMFSwerebuilt,MF0wasbuiltusingtheinformationof1HFanalysisand17LFanalyses,whileMF1wasbuiltwith5HFand17LFanalyses.ThesewerecomparedwithanHFSbuildusing17HFanalyses.ThecostoftheoptimizationusingMF0was13%andusingMF1was36%ofthecostoftheoptimizationusingtheHFS.Theobjectivefunctionwasincreased(thatismadepoorer)by5.47%and0.37%inperformanceforMF0andMF1,respectivelycomparedtotheobjectiveoftheHFoptimization.Eachmethodranfor7-10optimizationiterations.Theoverallairfoildesignwasimprovedby30%fortheMF0andby33%fortheMF1andHFwithrespecttothebaselinegeometry. Table3-1. Padronetal.,2016[ 1 ]optimizationcost,savingsandaccuracyreportasamodelforauthors.ItwasreportedthatasingleHFanalysiscosts15timesthecostofanLFanalysis.TheLFMerrorisabout18%comparedwiththeHFM.Theoptimizationtimewasreducedby87%withaperformanceof95.5%forMF0,andby64%withaperformanceof99.6%forMF1).Theoveralldesignwasimprovedby30% PropertyValueComments CostLF/CostHF0.07LF=Euler,HF=RANSErrorLF0.18HF=23,315points,LF=6,982pointsCostMFOpt./CostHFOpt.MF0=0.13,MF1=0.36MF0=1HF+17LF,MF1=5HF+17LFObjectivefunctionincreaseMF0=5.47%,MF1=0.37%W.r.ttheHFOpt. 65

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Inaddition,itwouldbeinformativetoincludetheaccuracyofLFM,HFMandMFMobtainedatthesamecomputationalcostandthecostoftheHFMandMFMobtainedforthesameaccuracy,ifpossible.Thisisdone,forexample,byPeherstroferetal.,2016b[ 106 ]whereinordertoaccountforaccuracyinthecalculationofthequantityofinterest,aplotispresentedgivingtherootmeansquareerror(RSME)asafunctionofthenumberofsamplesused.ThisanswersthequestionofhowaccurateistheMFMcomparedwithLFMandHFMatthesamecomputationalcost.Inaddition,asecondplotmayreporttimesavingsformultiplenumbersofsamplesoptions.ThisanswersthequestionofwhatarethesavingsassociatedwiththeimplementationofMFMcomparedwiththeHFsurrogateforthesameaccuracy. 3.8ConcludingRemarksMulti-delitysurrogates(MFS)havebeenapopulartopicofresearchinthepasttwodecades.SubstantialprogresshasbeenachievedinobtainingmoreaccurateMFS.Thispapernotesthattheuseofascalingparameterforthelow-delitydatainadditiontoanadditivecorrectionmayaccountforsomeofthisprogress.However,basedonasurveyofalargenumberofpapersitappearsthatresearchisstillneededtoprovideguidanceastowhenitisworthwhiletoinvesttheeortinusingMFS.FurtherresearchisneededtodeterminewhetherapplyingMFSisworthwhileforagivenproblemandtoselectaproperMFSframeworkfortheproblem.Thiseortentailsrunningtwoormoresetsofsimulationsofdierentdelitiesandselectingsurrogatesandamethodforcombiningthem.Therearesomeindicationsthattheanswermaybeassociatedwiththebumpinessofthelow-delityfunctionorthebumpinessofthedierencebetweenthelow-delityandhigh-delityfunctions.ItisalsorecommendedthatquantitativecomparisondatabetweenLFM,HFM,andMFMbereportedinthefuturepublications. 66

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CHAPTER4EARLYTIMEEVOLUTIONOFCIRCUMFERENTIALPERTURBATIONOFINITIALPARTICLEVOLUMEFRACTIONINEXPLOSIVECYLINDRICALMULTIPHASEDISPERSION 4.1SummaryDenselayersofsolidparticlessurroundingahigh-energyexplosivegenerateinstabilitiesatlatertimesafterdetonation.Conjecturesastothecauseoftheseinstabilitiesincludeinhomogeneitiesintheinitialdistributionofparticles.Toreducecomputationalcostweuselargerinitialimperfectionsthatdevelopintoinstabilitiesmorerapidly.Uptotrimodalazimuthalperturbationswereimposedintheinitialparticlevolumefractiondistributionwithinanannularbedsurroundingahigh-energyexplosiveinacylindricaltwo-dimensionalmultiphaseexplosion.Weobservetheinitialmodalperturbationsintheparticlevolumefractiontoimpacttheparticledistributionatlatertimesduetowhatcanbecalledasthechannelingeect.Thechannelingeectincreasestheangularvariationinthenetvolumeofparticlescontainedwithinradialsectorsofthedomain,andalsoincreasesthedierenceintheradialextentofparticledistributionwithinthedierentsectors.ThedeparturefromaxisymmetryintheparticledistributionwasmeasuredbyintroducinganL2metricdenotedbyFandanL1metricdenotedby.Thevariablesconsideredaretheparametersofatrimodalsinusoidalperturbation(amplitudes,wavelengths,andrelativephases).Wendthemetricdependenceontherelativephasesbetweentheperturbationmodesisnegligible.Ontheotherhand,themetricdependenceontheeectivewavenumberisstrongest.Weconcludethatunimodalperturbationsamplifybothmetricsthemost.WealsondthatFismostampliedatawavenumberof20whileatawavenumberof21.Butthewavenumberofthispeakmodecanbeexpectedtobedependentonthecircumferentialresolution. 67

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4.2BackgroundExperimentshaveshownthatdenselayersofsolidparticlessurroundingahighenergyexplosiveundergoinstabilitiesastheyradiallydispersefollowingthedetonation[ 2 { 4 ].Figure 1-1 showsacylindricalmultiphasedetonationexperimentperformedbyFrostetal.in2012whichpresentsaninitiallycylindricallysymmetricconguration.Herethecentralcylindricalchargeissurroundedbyanannularbedofnearlysphericalparticles.Atlatertimes,however,thissymmetryislostduetothedevelopmentofinstabilities.Themechanismsgoverningtheformationandgrowthoftheseparticleinstabilitiesarepoorlyknown,butmightdependonthenatureoftheparticles,thegeometryofthecharge,themassratioofexplosivetoparticles,imperfectionsinthecasingcontainingtheparticles,inhomogeneitiesintheinitialdistributionofparticles,stresschainswithintheparticlebedduringshockpropagation,andothercausesnotyetconsidered[ 242 ].However,themainsourceoftheinstabilitiesandtheirgrowth/amplicationmechanismsarestillbeingdebated.Thepossiblesourcesaremanyandthemechanismsofgrowthareoftencomplexandinteracting.Thereforethefocusofthisworkislimitedtoinvestigatingafewspecicaspectsregardingthegrowthofdeparturesfromaxisymmetryobservedintheexperiments.Forinstance,inthiswork,weassumethesourceoftheinitialperturbationtobeanon-axisymmetricparticledistributionanddonotexploreotherpossibilities.Ourfocuswillbeoncharacterizinghowthenatureoftheinitialperturbationinuencestheinstabilitygrowthatlatertimes.Previousresearch[ 5 { 7 ]indicatesthatsingleandbimodalazimuthalperturbationsoftheinitialPVFleaveasignatureintheparticlecloudfortimesontheorderofmillisecondsandmore,afterdetonation.Inthiswork,wehavechosentoimposeuptotrimodalperturbationsintheinitialparticlevolumefraction.Withanensembleofmorethan1,800simulationsthatcoverawiderangeofinitialperturbations,westudytheamplicationofdeparturefromaxisymmetryatlatertimesanditsdependenceonthenatureoftheinitialperturbation. 68

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A B CFigure4-1. DavidL.Frost.EvolutionintimeofacylindricalmultiphasedetonationusingPETNexplosivesurroundedbyglassparticles.17September2012.Quebec,Canada.Source:D.L.Frost,Y.Gregoire,O.Petel,S.Goroshin,andF.Zhang,Particlejetformationduringexplosivedispersalofsolidparticles,PhysicsofFluids,vol.24,no.9,p.91-109,2012.A)Initialtime,beforethedetonation.Thecongurationishighlyaxisymmetric;B)2:5mstimeafterdetonation.Instabilitiesbegintoformmakingthecongurationrelativelyaxisymmetric;C)5msafterthedetonation.Instabilitiesarehighlydevelopedandthedeparturefromaxisymmetryisevident. ItisbelievedthatthegrowthofinitialperturbationsthatisobservedintheexplosivedispersalofparticlesisatleastinpartduetoRayleigh-Taylor(RT)[ 25 ]andRichtmyer-Meshkov(RM)[ 26 27 ]instabilities[ 243 ].TheRTinstabilityoccurswhenaninterfacebetweenaheavyandlightuidacceleratesinthedirectionoftheheavyuid,orequivalentlywhentheinterfacedeceleratesinthedirectionofthelighteruid.TheRMinstabilityariseswhenashockwave,orotherimpulseacceleration,travelspastadensityinterface[ 244 ].Inbothcases,theinstabilitygrowthisintermsofvorticalstructuresthatappearduetothebaroclinicproductionofvorticity.Furthermore,intheidealsetupofaplanardensityinterfaceofinnitesimalthickness,theinitiallineargrowthisexponentialandthegrowthratescalesasthesquarerootofthewavenumberoftheperturbation.Thus,theshortwavelengthsaretheonestogrowmostrapidlyandthesizeofthemostampliedmodeislimitedbyviscousandotherdiusionalmechanisms.TheRTandRMinstabilitieshavethepotentialtorapidlygrowanyinitialperturbationpresentintheparticlebed.However,thepresenceofnite-sizedinertialparticles,thesupersonic 69

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backgroundowandthepresenceofothercompressibleowfeaturesmaycausethistoactasnon-classicalRTandRMinstabilities.ItisalsowellestablishedthatthegrowthrateoftheRTinstabilityshiftsfromanexponentialgrowninthelinearregimetoanalgebraicgrowthandsaturationinthenon-linearregimeoncetheperturbationhasgrownsuciently.Thistransitionfromlineartonon-linearbehaviortypicallyoccurswhentheamplitudebecomeslargerthanroughlyfortypercentoftheperturbationwavelength[ 28 ].TheRTinstabilityofsphericallyexpandinggasfrontsresultingfromasphericalshocktubeproblemhasbeenstudiedbothintheinviscidandviscousowregimesusinglinearstabilityanalysis[ 29 30 ].Theseresultshavealsobeenextendedtocylindricalsystems[ 31 ].Theyalsoshowedthatdespitetheaddedcomplexityoftheowresultingfromasphericalorcylindricalshocktube,thesimpletheoryofEpstein[ 32 ]isabletopredictboththeexponentialandalgebraicgrowthbehaviorsremarkablywell.Similarly,theroleofRMinstabilityforsingleandmultimodalperturbationhasbeenstudiedingreaterdetailforagasonlysystem[ 33 ].Ofmorerelevancetothepresentstudyisthelinearinstabilityanalysisperformedtoinvestigatetheinstabilityofradiallyexpandingparticulatefronts[ 30 31 ].Itiswellknownthatlargerinstabilitiesmaybeinuencedbythefragmentingofthecontainerwhichholdsthehigh-energyexplosive.However,instabilitiesalsoclearlyformintheabsenceofthesecasings[ 4 ].ToexploretheRMinstability,Ripleyetal.[ 4 ]performedanumericalstudywhichspeciedinitialperturbationsusingshallowellipticaldimplesintheedgeofacircularchargewithaspacingfrequencycorrespondingtothenumberofexperimentallyobservedinstabilities.Thisstudydemonstratedthattheinitialedgeperturbationandasimpleparticleinteractionmodelleadtotheformationofcoherentinstabilities.Xuetal.[ 34 ]showedusingmesoscalesimulationsthatthenumberofparticleinstabilitiesisdictatedbytheinitialparticlenumberintheinnerlayerattheexplosiveinterface.Inanattempttolinkthisndingtothemacroscopicexperimentalobservations,Ripleyetal.[ 4 ]interpretedtheinner-layerparticlesasparticlefragmentsofacasing 70

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betweentheexplosiveandpackedparticlebed.Theinstabilityinducedbythefragmentsfromtheinnerboundarydictatedthelate-timenumberofmajorinstabilitystructures[ 35 ].Unsteadyeectsarealsoknowntoplayasignicantroleinmultiphaseowsundercertainowconditions.Whetherornottheyhaveanimpactonthegrowthofthemixinglayerisanopenquestion[ 36 ]andisaphenomenonofinterestinthisstudy.Thefocusofthepaperistoconsideraverylargenumberofnumericalsimulationswheretheinitialmodalperturbationoftheparticledistributionwithintheannularbedissystematicallyvaried.Inordertocomparehowtheradiallyexpandingparticledispersiondepartsfromanaxisymmetricdistribution,werstidentifymetricsthatwillbeshowntoperforminarobustmanner.Twometricswerechosen,whichrepresentformsofL2andL1norms.Duetotherandomnatureofparticledistribution,thereisunavoidablestochasticvariationinquantitiessuchasparticlevolumefraction.Forexample,twodierentsimulationswithidenticalkeyparameters,andaredierentonlyintherandomlocationoftheinitialparticledistribution,areshowntoevolvedierently.Suchsimulationsareusedtoestablishtherandomnoiselevel,whichthenisusedtoidentifythekeyparametersoftheinitialperturbationwhosetrueinuenceismuchstrongerthanthenoiselevel.Weobserveaninterestingchannelinginstabilitythatcontributestothegrowthofdeparturefromaxisymmetry.ThisinstabilityisrelatedtoRTinstability,butoneinwhichtheentirelayerofparticlesseemtoparticipate(notjustthefront).Thispaperisorganizedasfollows:Section 4.3 hastwosubsections.Subsection 4.3.1 describesthecomputationalmodeling,liststhegoverningequationsused,andshowsthephysicalmodels,Subsection 4.3.2 describesthegeometryofthecomputationaldomain,andtheprobleminitialconditions.Section 4.4 describestheperturbationsimposedontheparticles.Section 4.5 liststhesimulationsrunforobtainingtheresultspresentedinthispaper.Section 4.6 introducesthephysicalphenomenonstudiedinthesesimulationsandcomparestheeectsofthephenomenaonunperturbedandperturbedcases.Inthis 71

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section,wealsoexplainindetailtheobservedchannelingeectandtheradialevolutionoftheshockandparticlesinthesimulations.Section 4.7 describesthemetricsusedtocomparethedeparturesfromaxisymmetryintheparticlebedsinthesimulationsanddiscussestheeectsofthedierentperturbationparametersonthesemetrics.Theimpactofnoiseinthesimulationsisalsodiscussedinthissection. 4.3NumericalMethods 4.3.1ComputationalModelingPhysicalexperimentsinvolvingtheexplosivedispersalofaparticlebedcancontainbillionsofparticles.Theseparticleshavediametersusuallyontheorderofmicrometerswhiletheoverallexplosiontakesplaceonalengthscaleofmeters.Duetothisdisparityinthemagnitudeofthelengthscales,itisimpracticalfromacomputationalviewpointtofullyresolvetheuidowaroundeachindividualparticle.However,wearealsointerestedintheevolutionoftheinstabilitiesintheparticlebedasthebeddispersesduetotheexplosion.Thismakesitcrucialtobeabletotrackthetrajectoriesofindividualparticlesthroughouttheow.Forthesereasons,thenumericalsimulationsdiscussedinthispaperwilltreattheuidphaseasacontinuumwhiletreatingtheparticlesasindividualpointmasses,resultinginanoverallEulerian-Lagrangianapproach. 4.3.1.1GoverningequationsFortheuidphase,weneglecttheeectsofviscosityandconductivity.Thus,thegoverningequationsarethecompressible,multiphaseEulerequations.Thuswesolvetheindividualequationsofcontinuity,momentum,andenergyexpressedinanEulerianframeworkasfollows @(gg) @t+r(ggug)=0(4-1) @(ggug) @t+r(ggugug)+rpg=)]TJ /F1 11.955 Tf 12.28 8.09 Td[(1 VXiiFgpi(4-2) 72

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@(ggEg) @t+r(ggEgug)+r(pgug)=)]TJ /F1 11.955 Tf 12.28 8.08 Td[(1 VXii(Ggpi+Qgpi):(4-3)Intheaboveequations,thesuperscriptgdenotesquantitiesassociatedwiththeuidphase.Specically,gisthegasdensity,gisthegasvolumefraction,ugisthegasvelocity,pgisthegaspressureandEgisthetotalenergyofthegas.Thetotalenergycanbedecomposedas Eg=eg+1 2ugug(4-4)withegbeingthespecicinternalenergyofthegas.Sinceitiscomputationallyexpensivetosimulateeachindividualparticlewhichexistsinthesimulations,theconceptofacomputationalparticleisemployedinthiswork.Acomputationalparticlerepresentsagroupofphysicalparticleswiththeiraveragepositions,velocities,andtemperatures.Thenumberofphysicalparticlescontainedineachcomputationalparticleisgivenbyi.Theseequationsaresolvedusinganitevolumeapproach.TheconvectiveuxesarecomputedusingasecondorderaccurateAUSM+upscheme[ 245 ].ThecalculatedgradientsaremodiedbyusingaWENOreconstructionmethod[ 246 ].ThetimeintegrationisperformedwithathirdorderaccurateRunge-Kuttascheme.Thecodewhichimplementsthisuidsolverhasbeenpreviouslytestedandvalidatedforanumberofnumericalsimulationsofcompressibleowsinvolvingshockwaves[ 247 { 249 ].Thegoverningequationsforthelocation,velocity,andtemperatureofeachparticlearegivenbelow dxpi dt=upi(4-5) mpidupi dt=Fgpi+Fppi(4-6) mpiCpidTpi dt=Qgpi:(4-7) 73

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Intheseequations,thesuperscriptpdenotesquantitiesassociatedwiththeparticlephase.Thevariablesxpi,upiandTpiarethepositionvector,velocityvectorandtemperatureoftheithcomputationalparticle.ThequantitiesFgpiandQgpiarethemomentumandheattransferredtoeachparticlefromthebackgrounduidow.Intheabovemomentumequation,Fppiistheforceontheithparticleduetointerparticlecollisionswithallotherparticles.InEq.( 4-3 ),Ggpidenotestheamountofkineticplusinternalenergytransferedtotheuidduetothehydrodynamicforcesontheithparticle.ThemodelsforthesetermswillbediscussedinSection 4.3.1.2 4.3.1.2Physicalmodels ParticleforcemodelInthiswork,boththesteadyandunsteadyforceswhichimpacttheparticlefromtheuidareincorporated.Thetotalforceimpartedonasingleparticlefromtheuidcanbedecomposedasfollows: Fgpi=Fqs+Fpg+Fiu+Fvu(4-8)wheretheindividualforcetermsontherighthandsiderepresentthequasi-steady,pressuregradient,inviscid-unsteadyandviscous-unsteady(orBassethistory)forcetermsasdenedin[ 250 ],[ 251 ]and[ 252 ]respectively.Thisdecompositionwasoriginallyderivedinanincompressibleregime.Thisformisalsoapplicableinthecompressibleregimeofthisworkifappropriatemodicationsaremade.Thesemodicationscomebyapplyingcorrectiontermstothequasi-steady,inviscid-unsteadyandviscous-unsteadyforcesinordertoaccountforniteReynoldsnumber[ 253 ],niteMachnumber[ 250 251 ]andvolumefraction[ 254 ]eects.Takingtheseintoaccount,eachoftheforcecomponentsareexpressedas: Fi;qs=3gidpi(ugi)]TJ /F8 11.955 Tf 11.95 0 Td[(upi)Cd(Repi;Mpi;pi)(4-9) 74

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Fi;pg=)]TJ /F4 11.955 Tf 10.49 8.09 Td[((dpi)3 6(rP)i(4-10) Fi;iu=)]TJ /F4 11.955 Tf 10.49 8.09 Td[((dpi)3 6Cm(Mpi;pi)D(giugi) Dt+d(giupi) dt(4-11) Fi;vu=3gidpiCvu(pi)ZtKvu(t)]TJ /F4 11.955 Tf 11.96 0 Td[(;Repi;Mpi)(rP)i+d(giupi) dtt=d:(4-12)Intheabove,theuidquantitieswithasubscriptiaretobeinterpretedastheuidpropertyevaluatedatthepositionoftheithparticle.Also,theparticleReynoldsandMachnumberaredenedas: Repi=dpijugi)]TJ /F8 11.955 Tf 11.96 0 Td[(upij gi;Mpi=jugi)]TJ /F8 11.955 Tf 11.96 0 Td[(upij agi;(4-13)wheregiandgiarethekinematicanddynamicviscositiesoftheuidandagiistheambientspeedofsoundoftheuid,bothevaluatedattheparticleposition.Also,dpiisthediameterofthecomputationalparticle.ThespecicexpressionsforthecorrectiontermsCD,CmandCvuandfortheviscous-unsteadykernel,Kvu,arediscussedindetailinLinget.al[ 255 ]. ParticlecollisionmodelThemodeltoaccountforthecollisionsamongparticlesistakenfromtheinterparticleforceintheLagrangianframeworkgivenby[ 256 ]as Fppi=)]TJ /F1 11.955 Tf 18.81 8.09 Td[(1 pipirpp;i;(4-14)wheretheforceiscomputedasthegradientofthecollisionalpressureactingontheparticleasisdoneinEulerian-Eulerianmultiphasesimulations.Theisotropicstresstensorusedinthisworkisgivenby[ 257 ] pp;i=Ps(pi) pcp)]TJ /F4 11.955 Tf 11.96 0 Td[(pi:(4-15) 75

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Intheabove,pcpistherandomclose-packingvolumefractionlimit.Forthemonodispersed,sphericalparticlesinthesimulations,thisvalueistypically0:63.FollowingtheworkofAnnamalaiet.al[ 5 ],thevaluesforPsandaresetto107Paand3respectively. ParticleheattransfermodelSimilartotheexpressionfortheforcetransferredtotheparticlefromtheuid,theheattransferredtotheparticlefromtheuidcanbedecomposedintoquasi-steady,undisturbed-unsteadyanddiusive-unsteadycomponentsas Qgpi=Qqs;i+Quu;i+Qdu;i:(4-16)Theundisturbed-unsteadytermissettozerointhiswork.Theexpressionsfortheothertwocomponentsare Qqs;i=dpigiNu(Tgi)]TJ /F4 11.955 Tf 11.95 0 Td[(Tpi)(4-17) Qdu;i=2dpiq giCgpgiZtKqvu(t)]TJ /F4 11.955 Tf 11.96 0 Td[()DTgi Dt)]TJ /F4 11.955 Tf 13.15 8.08 Td[(dTpi dtt=d(4-18)wheregiisthethermalconductivityofthegas,NuistheNusseltnumberandCgpisthespecicheatoftheuidatconstantpressure.Theformforthediusive-unsteadykernel,Kqvuisgivenin[ 258 ].ItshouldbenotedthatthereisnoextensionforthekernelforniteMachnumberssocompressibilityeectsareneglectedforthisparticularheattransfercomponent. EquationofstateTheequationofstateusedtoclosethesystemofuidgoverningequationsisgivenbytheidealgasequationofstate.Givengandeg,theseyieldthegaspressureandtemperatureas pg=()]TJ /F1 11.955 Tf 11.96 0 Td[(1)geg=gRTg(4-19) 76

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whereRisthespecicgasconstantforagasandistheratioofthespecicheatsatconstantpressureandvolume.Thesimulationsinthisworkwillassumetheentireambientuidphasetobeairwhichhasvaluesof1.4and287.04J kgKforandR,respectively. 4.3.2ProblemDescriptionThecomputationaldomainisatwo-dimensionalsliceofacylinderofdiameter1:20mtakenperpendiculartothecylinderaxis.Thiswaschosentocontaintheblastwaveduringtheentiresimulationtimeof500s.Thisdomainiscomprisedofa0:0076mdiameterinnercirclecontaininghot,high-pressuregaswhichissurroundedbyanannularparticlebedofouterradius0:05m.Theremainderofthedomaincontainsambientair.Exceptfortheinnercircle,whichcontainsthehigh-pressuregassimulatingtheproductsofdetonationofaninitiallyhigh-energycharge,therestofthedomainisinitiallyunderstandardconditionsofpressureandtemperature.Figure 4-2 showsaschematicofthecomputationaldomain. Figure4-2. Schematicofthecomputationaldomain(nottoscale). Theinitialconditionsforthegasphaseareasfollows.Thehigh-energyexplosiveinthecharge(i.e.r0:038m)istakentoberepresentativeofpentaerythritoltetranitrate(PETN).Usingdatatakenfrom[ 259 ],thegasinsideofthechargeissettoaninitial 77

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densityofg=1770kg m3andanenergycontentofgEg=10:089GJ m3atzerovelocity.Outsideofthecharge,thegasisinitializedatstandardatmosphericconditionsofg=1:203kg m3andpg=101325Paatzerovelocity.Forthiswork,theentiregasphaseisgovernedbytheidealgasequationsandallremainingnecessarygasvaluesarebackedoutofEq.( 4-19 )usingtheconstantsforairgiveninSection 4.3.1.2 .Thepropertiesfortheparticlephaseattheinitialtimeareasfollows.Theparticlesaretakentobemadeentirelyofglasswithadensityofp=2500kg m3andadiameterof100m.TheheatcapacityoftheparticlesisCpi=450J kgK.Theparticlesareinitiallytakentooccupyavolumefractionof5%intheannularregion0:0038mr0:05m.Thenumberofcomputationalparticlesinallofthesimulationsis31;250,whichisrandomlydistributedwithintheannularregionwithuniformprobability.Withineachnitevolumecellthesuperparticleloadingfactor,,oftheparticlesinsidethecellisadjustedsuchthatthecellaveragedPVFwithinthecellequalsthedesiredPVF.ismaintainedconstantthroughoutthelifeoftheparticle.Theratioofthemassoftheparticlebedtotheinitialmassofthechargeis17:9.Thecomputationaldomainisdividedintotworegions.A6464cellCartesianmeshisforcedintotheinnercircleofthedomainwhichinitiallycontainsthehigh-pressuregas.Therestofthedomainusesapolarmeshwith125and256cellsintheradialandazimuthaldirections,respectively.Figure 4-3 isaschematicofthegrid.Thegridshowninthegureismuchcoarserthantheactualgridandisshownforillustrationpurposesonly.Theresolutionemployedinthisstudyismodestsinceweplantoconsiderandcomparetheresultsofmorethan1,800simulations.However,wehaveperformedsimulationsusingnergridsinordertoestablishtheadequacyofthecoarsergridforthepresentproblem.InFigure 4-4A weshowtheconvergenceoftheshocklocationforthethreegridsdescribedinTable 4-1 .Itcanbeobservedthattheshocktrajectoryresultsarenearlyindependentofthegridresolution.InFigure 4-4B weshowtheupstreamparticlefrontlocationasafunctionoftimeforthethreedierentresolutions.Theconvergence 78

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A B CFigure4-3. Schematicofthegridsetupusedforillustrationpurposesonly.Thegridinthispictureiscoarserthantheactualgridforeasiervisualization.NotethattheoutergridispolaranditsangulardivisionsaredeterminedbythenumberofCartesiancellsoftheinnergrid.Theradialdivisionsaresetindependently.A)Entiregrid,0x;y0:6m;B)Zoomedgrid,0x;y0:06m.Theouterpolargridcanbeclearlyseen;C)5msafterthedetonation.Zoomedgrid,0x;y0:01m.TheinnerCartesiangridcanbeclearlyseen. isnon-monotonicandtime-dependent.AlsoasnotedinFigure 4-22 ,duetothechaoticmotionoftheparticles,thereisstatisticalvariationinthelocationofupstreamanddownstreammostparticlebetweendierentrunsofthesameresolution.Basedontheseresults,andtheneedforalargenumberofsimulations,thelow-resolutionresultsareadequate.Furthermore,theuncertainty(orthenoiselevel)intheparticlemetricstobeusedintheanalysiswillbequantied. Table4-1. Griddetails GridNameRadialDivisionsAzimuthalDivisionsTotalNumberofCellsComputationalParticles LF12525636,09631,250MF250512144,384125,000HF5001024577,536500,000 Figure 4-5 showstherobustnessofthemetricdespitethegriddiscretization.InFigure 4-5A themetricFextractedfromthelow-delity(LF)gridisplottedasafunctionoftheeectivewavenumber,keff.ThedatafromtheLFgridistheoneusedinthispaper.InFigure 4-5B themetricFextractedfromthehigh-delity(HF)gridisplottedasafunctionofkeff.Inthegures,bluedotsrepresentcaseswithtrimodalinitial 79

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A BeFigure4-4. SimulationmetricsasafunctionoftimeforthethreedierentgridsdescribedinTable 4-1 .A)Shocklocationasafunctionoftime;B)Upstreamfrontofparticlesasafunctionoftime. perturbationsintheparticlebed,whileorangedotsrepresentcaseswithsinglemodalinitialperturbations.Inbothgures,theleadingperturbationsareunimodal.Therefore,themaximumamplicationofthedeparturefromaxisymmetryisfoundtobeduetounimodalperturbationsindependentlyofthegrid.Ontheotherhand,thewavenumberthatmaximizesFfortheunimodalinitialperturbationseemstobegriddependent.WiththeLFgrid,wehaveobtainedanoptimalwavenumberof20,whiletheHFgridresultsshowanoptimalwavenumberof12. 4.4PerturbationMethodologyItisbelievedthatinstabilitiesobservedintheexplosivedispersalofparticlescanbetracedbacktotheinitialdisturbancespresentinthegasand/orparticulatephases[ 3 ].Previousresearchindicatesthatinitialperturbationsinthehothigh-pressuregasshownonoticeabledierencewiththeunperturbedcaseatlatertimesindicatingthattheeectofaninitialperturbationinthedetonationproductsisnotimportant.Incontrast,aninitialperturbationinthebedofparticlesintermsofinitialvolumefractiondistribution 80

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ALFData BHFDataFigure4-5. ThemetricFasafunctionofkeff.Bluedotsrepresentsthecaseswithatrimodalperturbationwhiletheorangedotsthecaseswithasinglemodalperturbation isampliedoverlongerperiodsoftime[ 5 ].Thisperhapscanbeexplainedduetothefactthatthemassoftheparticlesisnearly17:9timeslargerthanthatoftheexplosiveproducts.Inthispaper,wegoastepfurtherandinvestigatewhichkindofperturbationsintheinitialPVFgrowthemostatlatertimes.ThePVFisdenedasthevolumeofparticlesinacomputationalcelldividedbythevolumeofthecell.AninitialbasePVFof5%,whichisrelativelylow,ischosentoavoidtheeectsofcompactionintheparticlebed.TheperturbationsimposedinthePVFareinspiredby[ 5 ].ThebasePVFwasperturbedusingasuperpositionofuptothreesinusoidalwavesinthecircumferentialdirection.ThethreemodeperturbationtothePVFcanbeexpressedas p()=p0[1+A1cos(k1+1)+A2cos(k2+2)+A3cos(k3+3)];(4-20)wherep0istheuniformbaselinePVFthatappliestotheentirebed.HereeachmodeisdenotedbyitsamplitudeA,wavenumberkandthephase.Duetocylindricalsymmetry,andwithoutlossofgenerality,wecanset1=0andthus,onlythephaseoftheothertwomodeswithrespecttotherstmodedeterminesthePVFperturbation.Notethattheaboveperturbationisonlyalongthecircumferentialdirectionandthereforepis 81

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afunctiononlyofandnotoftheradialcoordinate.Inthenumericalsimulations,the31;250computationalparticlesarerandomlydistributedwithineachcellintheannularbedandthesuperparticleloadingisdeterminedaccordingtothevolumefractionvariationgivenabove.TherandomlychosenlocationofparticleswithinthebedintroducesadditionalhighwavenumbeructuationsinthePVFontopofthethreemodeperturbationexplicitlyintroducedintheinitialPVF.Inthisworkwewilldenetheunperturbedsimulationasonewheretheinitialpdistributionisuniform(i.e.,A1=A2=A3=0).Wedenetheunimodalperturbationtoincludeonlyonemode(A1>0;A2=0;A3=0)anditischaracterizedbythetwoparametersA1andk1.Thebimodalpertrbationisdenedtoincludetwomodes(A1>0;A2>0;A3=0)anditischaracterizedbythesetofveparametersfA1;k1;A2;k2;2g.Finallythetrimodalperturbationincludesallthreemodes(A1>0;A2>0;A3>0)andischaracterizedbyeightparameters:fA1;k1;A2;k2;2;A3;k3;3g.Higherthantrimodalperturbationsarenotconsideredinthiswork,partlyduetothefurtherincreaseinthenumberofassociatedparametersandpartlyduetothefactthattherandomdistributionofparticlesanywayintroducesadditionalhiddenmodesofperturbation.Aswecomparethegrowthoftheseperturbationsfromtheirinitialvalueinordertodeterminethecombinationofmodesthataremostlikelytobeampliedbytheinstability,itisimportanttoperformsuchacomparisononafairbasis.Wewillnormalizetheperturbationamplitudeatanylatertimewithitsinitialvalueandtheratiowillprovideanobjectivemeasureofinstabilitygrowthfactor.Inaddition,ithasbeenwellrecognizedthatthegrowthrateofhydrodynamicinstabilitiesstronglydependsontheperturbationamplitude,withthegrowthratebeingexponentialatinnitesimalamplitudesoftheperturbationandthegrowthratetypicallyslowingdownathigherniteamplitudesduetononlineareects.Thus,forafaircomparisonofthedierentperturbationmodes,wesettheinitialtotalperturbationamplitudetoaconstant.Inotherwords,weimposethe 82

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condition A21+A22+A23=constant;(4-21)andinthisworkwesettheconstanttobe0.02,yieldinga14%totalamplitude[ 6 ].ThisconstraintonthemeansquareperturbationintheinitialPVFreducesthenumberofparametersbyoneintheunimodal,bimodalandtrimodalcases.Toillustratethenatureoftheinitialperturbations,Figure 4-6 comparestheinitialPVFcontoursfortheunimodalcasewithacasewhereatrimodalperturbationisimposed.Theincreasedcomplexityoftheinitialdistributionofparticlesisclear.OntopofthePVFcontours,weshowwithacontinuousblacklinetheparticlevolume(PV)proleasafunctionof.Notethat,duetotheindependenceontheradialcoordinatebetween0:0078m
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A BFigure4-6. PVFcontoursattheinitialtime.PVFintheradialcoordinateisinitiallyconstantbetween0:0078m
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Table4-2. Casessimulated.Number,shortdescriptionandparametersofthesimulationsperformedforthispaper.Notation:*:randomgeneratednumber,C:MustsatisfytheconstraintofEq.( 4-21 ),R:Realnumber,I:Integer,theintervalsindicatethedomainboundarieswheretheparametercanbesampledfrom Numberofsimulations Shortdescription A1 A2 A3 k1 k2 k3 1 2 3 1 Unperturbed 0 25 Unimodal p (0:02) I[1;25] 0 100 Bimodal CR(0;p (0:02)] I[1;25] 0 R[0;2] 1,600 Trimodal CR(0;p (0:02)] I[1;25] 0 R[0;2] 20 Seedvariation 0.0034 0.1413 0.0061 14 1 15 0 3.7623 2.2688 20 0.0139 0.0767 0.1180 5 1 24 0 4.0835 0.7328 20 0.1359 0.03695 0.0134 20 16 22 0 2.8835 4.2303 20 Phasevariation 0.0034 0.1413 0.0061 14 1 15 0 R[0;2] 20 0.0139 0.0767 0.1180 5 1 24 0 R[0;2] 20 0.1359 0.03695 0.0134 20 16 22 0 R[0;2] 1,846 85

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4.6Results:InstabilityCharacterization 4.6.1UnperturbedCaseAlthoughcomputationalparticlesarelocatedrandomlyincomputationalcells,thePVFpercellismaintainedconstantinthiscase,thereforewerefertothiscaseastheunperturbedcase.Fig. 4-7 showsthegasdensitycontoursandthePVFcontoursatthreedierenttimes.Thiscylindricalshock-tubeproblemischaracterizedbyseveralcompressibleowfeatures,althoughallofthemarenotimmediatelyvisibleinFig. 4-7 .Themostimportantamongthemistheprimaryshockthatpropagatesradiallyoutandhasbeenmarked(PS)inFig. 4-7C .Att=25stheprimaryshockescapesofthebedofparticlesandcontinuestopropagateoutintotheambient.ThepositionoftheprimaryshockasafunctionoftimeisgiveninFig. 4-7D .Thesecondfeaturetoobserveisthecontactinterface(CI)thatseparatestheproductsofdetonationfromtheambientair.TheCIalsoinitiallypropagatesthroughthebedofparticlesandattimet=50sreachestheouteredgeoftheparticlebed.UnlikePS,theCIdoesnotpropagateoutforeverbutreachesanasymptoticradius(Fig. 4-7D )[ 36 260 261 ].WhilePSandCIpropagateout,anexpansionfanpropagatesinwardintothehothigh-pressuregasthatisinsideoftheparticlebed.Theheadoftheexpansionpropagatesinwardtowardstheaxisofthecylinderatthelocalspeedofsoundandreectsotheaxis.Meanwhile,theinitiallyoutwardpropagatingtailoftheexpansionbecomesasecondshockandturnsaroundtocollapseinwardtowardstheaxis.Thehigh-pressuregasintheinnercoreundergoesover-expansionandthepressureintheinnerregion,infact,goesbelowtheambientpressurewithevenaregionofinwardowtowardstheaxis.Theabovescenariohasbeenobservedforsinglephasecylindricalexpansionows.Inthepresentcase,theowgetsfurthercomplicatedduetothepresenceoftheparticles.WeobservethatPSandtheCIarebarelyaectedbythepresenceofparticlesbecauseofthelowerPVF,thereforetheirbehaviorwithorwithoutparticlesispracticallythesame.However,severaladditionalfeaturescanbeobserved.ThePShitstheinnerparticlefront 86

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(IPF)rst,thereforetheIPFgainsmomentumwhiletheouterparticlefront(OPF)isyetstill.IPFandOPFaveragedlocationaremarkedinFig. 4-7C .Thisscenarioiswhatwecallthecompactionphase.AfterthePSgetsoutofthebedofparticles,theparticlesexpandintheradialdirectionmarkedbytheIPFthatmovesoutradiallyataslowervelocitythantheOPF.Thisindicatesthattheaccelerationand,accordingly,theradialspeedofparticlesismuchhigherattheOPFthanattheIPF.ThetimehistoryofIPFandOPFarealsopresentedinFig. 4-7D ,notethattheyhavebeendeterminedusingacutoof5%withrespecttothepeakPVFatthattime.Duetoitshigherinertia(particledensityisaround700timesthedensityofthehigh-pressuregasatthistime),theparticlesaresluggishandaccelerateslowly,sotheirvelocityincreasesatamuchslowerrate.Ontheotherhand,atlatertimeswhilethegasvelocitycontinuestodecreaseduetoradialexpansion,theparticlevelocitycontinuestoremainsignicant,againduetoitsinertia.Anotherimportantfeaturetobeobservedisthewave-likedistributionoftheparticleswhereregionsofconcentratedparticlesareseparatedbyvoidregionsofmuchlowerparticleconcentration.Suchsegregationorlayeringofparticleshasbeenwidelyreportedinuidizedbeds[REFs]andparticlebedssubjectedtoexpansionows[REFs].Volumefractiondependentdragontheparticlehasgenerallybeenconsideredthesourceofsuchinstability,whichappearstohaveastronginuenceinthepresentproblem.Thoughtheactualparticledistributionisrandom,acircumferentiallyaveragedPVFcanbecalculatedandplottedasafunctionofradius.Figure 4-8 showsthetemperatureandpressureofthegasatthenaltime(500s).AlsoshownaretheparticleradialvelocityandPVFcontoursat500s.Thehighestgaspressureisfoundneartheshockwaveanditisnotalteredbythepresenceoftheparticlessubstantiallywhencomparedtotheresultofthecorrespondingowwithouttheparticles.Thissmalleectofparticlesinthepresentcaseisduetothemodest5%initialPVF.Themaximumpressuredropsfromtheinitial4GPa(initialpressureinthecharge)to40MPa.Ontheotherhand,thetemperaturereachesitsmaximumwherethe 87

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A B C DFigure4-7. Evolutionofgasdensity,PVF,PS,CI,IPF,andOPFfortheunperturbedcase(A1=A2=A3=0).A)PVFcontours(right)andgasdensitycontours(left)att=0;B)PVFcontours(right)andgasdensitycontours(left)att=250s;C)PVFcontours(right)andgasdensitycontours(left)att=500s;D)Primaryshock(PS),contactinterface(CI),innerparticlefront(IPF),andouterparticlefront(OPF)averagedradiallocationasafunctionoftime. 88

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bulkofparticlesislocated.ThepeakvalueofPVFhasdoubleditsinitialvalueduetocompactioneects.WhilethePVFinthebulkofparticlesthatmarktheIPFishigh,thePVFoftheoutermostparticlesisquitelow.Wehavesetthecuto5%ofthepeakPVFvalueinthegures,whichcorrespondstomorethananorderofmagnitudereductioninPVF.TheseouterparticlestravelaboutthreetimesfasterthanthebulkofparticlesneartheIPFat500s.Someparticlesaretravelingwiththeshock.ThesecanbeseenintheparticlevelocityquadrantofFig. 4-8 buttheirPVFisnegligibleandtheyarenotconsideredaspartofthebulkofparticles. Figure4-8. UnperturbedinitialPVF.Gaspressure,gastemperature,particlevelocity,andPVFcontoursfort=500s. 4.6.2PerturbedcasesInthissection,weshowsimilarresultsasinSection 4.6.1 butforacaseinwhichinitialPVFwasperturbed.Aunimodalperturbationcasedenedbythesingleparameterk1=10willberstconsidered.Fig. 4-9 showsthetimeandevolutionofgasdensity 89

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contoursandthelocationofthecomputationalparticlesatthreedierenttimes.Onaveragethegasdensityshowsthesamebehaviorasintheunperturbedcase.ThePS,inparticular,remainscylindricaldespitetheunimodalperturbationintroducedintheinitialPVF.AcarefulexaminationofthegaspropertieswillshowthatthisisnotthecaseawayfromthePS.ShowninFigure 4-10 arethetemperatureandpressureofthegasandtheparticlevelocityandPVFat500s.TheCIandtheexpansionfanthatpropagateintotheinitialhigh-pressuregasarebeingimpactedbytheinitialunimodalperturbation.ThemodalperturbationpatterncanbeseeninthegasdensityclosetotheorigininsidetheIPF.Incomparison,thegaspropertiesintheregionoccupiedbytheparticlesarenotonlyinuencedbytheinitialmodalperturbationbutalsoareinuencedbytherandomdistributionofparticlesthatgiverisetohighwavenumberoscillations.Thus,theinitialperturbationintroducedinthedistributionofparticleshasextendedintothegasproperties.TheoverallbehavioroftheparticlesisalsosimilartoFig. 4-8 withthedierencethatsmallinitialmodalperturbationsinthePVFleavetheirsignatureatlatertimesimprintinghighlycorrelatedmodalinstabilitiesintheparticledistribution.Acleartenmodewavepatterncanbeobservedintheparticlebed.Thismodaldistributionextendsfromtheinnertotheouterparticlefront.However,theamplitudeofthewavyparticlepositionismuchlargerattheOFPwheretheparticleconcentrationisdilutethanattheIFPwherethePVFisstillclosetotheinitialPVF.Oneimportanteectoftheperturbationisthatwecanobservetheoutermostedgeoftheparticlesextendsfartheroutandalmostextendsuptotheshockfront.ThisisclearlyseeninFig. 4-10 inthequadrantwheretheparticlevelocityisshown.ItisthisinstabilityandthegrowthofthewavyundulationinboththePVFandtheradiallocationoftheparticleswhichisofinteresttothepresentwork.Wewillconsiderthedependenceofthisinstabilityonthenatureoftheinitialmodalperturbation. 90

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A B C DFigure4-9. Evolutionofgasdensity,PVF,PS,CI,IPF,andOPFfortheunimodalperturbedcasewithparametersA1=p 0:02;A2=A3=0;k1=10;1=0.A)PVFcontours(right)andgasdensitycontours(left)att=0;B)PVFcontours(right)andgasdensitycontours(left)att=250s;C)PVFcontours(right)andgasdensitycontours(left)att=500s;D)Primaryshock(PS),contactinterface(CI),innerparticlefront(IPF),andouterparticlefront(OPF)averagedradiallocationasafunctionoftime. 91

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Figure4-10. PerturbedinitialPVF.ThecaseconsideredistheunimodalcasewithparametersA1=p 0:02;A2=A3=0;k1=10;1=0.Gaspressure,gastemperature,particlevelocity,andPVFcontoursfort=500s. AlthoughbylookingattheazimuthallyaveragedquantitiesinFigures 4-7D (unperturbedcase)and 4-9D (perturbedcase)wedonotseesubstantialdierences.BycomparingFigure6(unperturbedcase)andFigure 4-10 (perturbedcase)weseethat,whiletheshockwasnotaectedbytheperturbationsinparticles,theyleadtoamuchhigherparticledeparturefromaxisymmetry.ThisphenomenonisdiscussedindetailinSection 4.6.3 4.6.3ChannelingEectAnimportantaspectoftheparticledistributioninFigs. 4-9 and 4-10 willbeconsidered.NotethattheinitialmodalperturbationwasinPVF,whichvariedinthe-direction.Theradialextentoftheparticlesatt=0wasindependentofthe-direction(i.e.,theIPFandOPFwereatr=0:038andr=0:5manddidnotvaryinthe 92

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circumferentialdirection).Yet,atlatertimesinFigs. 4-9 and 4-10 theparticlepositionovertheentirebedhasdevelopedanoscillation.Thoughnoteasilyapparentinthesegures,thePVFalsoshowsthetenwaveoscillationinthecircumferentialdirectionandthisoscillationperfectlycorrelateswiththeinitialmodalperturbation,andtheamplitudeoftheoscillationshasgrownovertime.ThetranslationoftheinitialPVFperturbationintoparticleradialpositionperturbationatlatertimesisduetothechannelingeect.ShownontheouterperipheryofFig. 4-11 isthepatternofinitialPVF,wherethecrestsindicateradialsectorsofhigherthanaveragePVandtroughsindicateradialsectorsoflowerthanaveragePV.Itcanbeclearlyobservedthattheentirebedofparticles(fromIPFtoOPF)hasmovedradiallyfartheroutintheradialsectorswheretheinitialPVFwaslow,andinradialsectorswheretheinitialPVFwashightheentirebedofparticlesisobservedtolagbehind.Thiscanbeexplainedbythefactthatthegasndsapreferentialpathintheradialsectorswherefewerparticlesarelocated,i.e.wherethePVFislower.Attheverybeginningfollowingthereleaseofhigh-pressuregas,thegasimmediatelyacceleratesforward,whiletheinitialdistributionofparticlesremainsrelativelyfrozen.Inthislimit,theowcanbeassumedtoapproximateaporousmediaow.TheradialsectorswithlargerPVFwillappearasporousnozzleswithalargerareacontraction,whilesectorswithsmallerPVFwillappearasporousnozzleswithasmallerareacontraction.Thepost-shockowinthepresentproblemissupersonic,thereforethenozzlingeectistodecreasethevelocityinregionsoflargervolumefraction.Thisreductionisfurtherstrengthenedbytheincreaseddragonthegasowfromthehigherconcentrationofparticles.TheparticleslocatedinthesesectorsalsogainmoremomentumthanparticleslocatedinradialsectorswithhigherPVF.ThefastermovingparticlesintheradialsectorsoflowerPVFnotonlymovefartheraheadintheradialdirectionbutalsocircumferentiallymigrateintoneighboringsectorsofhigherPVF.Thiscircumferentialmigrationispartlyduetointer-particlecollisionandpartlyduetothegas-mediatedinteractionbetweenneighboringparticles.Irrespective, 93

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theresultofthislateralmigrationistoincreasethenumberofparticlesinthehighPVFsectorsattheexpenseofsectorsoflowerPVF.This,inturn,willincreasetheradialvelocityandradiallocationofparticlesinthelowPVFsectors,furtherincreasingthecircumferentialmigration.ThiscyclecanbeexpectedtocontinuecontributingtobothanincreaseinthedierencebetweenthePVFinthehighandlowPVFsections,andalsotoanincreasingdierencebetweentheradialpropagationofparticlesinthesesectors.Duetothefeedbackeectexplainedabove,thismechanismwillbetermedchannelinginstability.Summarizingtheeectsofthechannelinginstabilityareto(i)increasethesectortosectorvariationinthenetvolumeofparticlescontainedwithinthesesectors,and(ii)toincreasethedierenceintheradiallocationofparticleswithinthedierentsectors.Boththesemechanismsappearasdeparturesfrominitialaxisymmetryinparticlelocation.ThisinstabilitymechanismcanbeaddressedwithrespecttotheRayleighTaylor(RT)instabilityprocess.Ifweconsidertheparticle-ladengasasaheaviermixture,thentheIPFcanbeconsideredasdensityfrontacrosswhichthemixturedensityincreaseswithincreasingrandOPFisadensityfrontacrosswhichdensitydecreaseswithincreasingr.Inthiscase,iftheIPFacceleratesradiallyoutthenitissubjectedtoRTinstability,whiletheOPFwillundergoRTinstabilitywhenitdecelerates.Inbothcases,theinitialperturbationfortheinstabilitycanarisefromtheinitialPVF.SuchRTinstabilityofradiallyexpandinggasandparticulatefrontshavebeenconsideredbyMankbadiandBalachandar,2014[ 260 ].Inthepresentproblem,asseeninFigs. 4-7D and 4-9D ,theIFPacceleratesrstwhiletheOFPhasstillzerovelocity.TheOFPwillstartmovingasitispushedbytheparticles.WhentheOFPgeneratesoscillations,theycanbeclearlyseenontherightsideofFig. 4-11 wherethePVFisshown.NotethatforFigs. 4-7 4-8 4-9 ,and 4-10 thecolorbarrangewassetfrom0.2%to10%.InFig. 4-11 weallowedthePVFrangetovaryfrom0.01%to2%makingtheoscillationsdescribedbeforeclearlyvisible. 94

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Figure4-11. Gasvelocity(left)andPVFcontours(right)atnaltime,500s.ThecaseconsideredistheunimodalcasewithparametersA1=p 0:02;A2=A3=0;k1=10;1=0.TheoutercontourrepresentstheinitialPV.Becauseofthechannelingeect,thegasmovesfasterinsectorswherethePVwasinitiallylower. Wenowconsiderthegrowthandthenatureoftheinstabilityofsomeofthemultimodalperturbationcasesinordertoreinforcethepointspresentedabove.Inparticular,weconsidertwoextremecases:therstbeingtheoneinwhichtheinstabilitygrowthovertimewasmaximalandthesecondbeingthecasewheretheinstabilitygrowthwastheminimum.Bothcasespresentnon-axisymmetricinitialPVFperturbationswithkeff=10.Thiswillallowforacomparisonofthenatureofinstabilityinthesetwoextremecases.InFigure 4-12 wepresentthecaseofmaximuminstabilitygrowthandthiscasehasaninitialtrimodalperturbationwithparametersA1=0:039145;A2=0:116199;A3=0:070466;k1=1;k2=12;k3=8;2=6:000171;3=0:378179.Frame( 4-12A )showscontoursofgasvelocity,whileframe( 4-12B )showstheparticlelocation 95

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coloredbyPVF.Alsoplottedintheframes 4-12A and 4-12B ontheouterperipheryaretheinitialPVproles.Clearly,thesector-to-sectorvolumevariationhassubstantiallyincreasedfromitsinitialvalueindicatingthestronginstabilityofthisinitialtrimodalperturbation.Theeectofthisinstabilityisalsofeltinthelargevariationintheradiallocationoftheoutermostextentoftheparticledistributionasafunctionof.ThisvariationintheradialextentoftheparticlebedismostpronouncedonlyattheOPFandisquitesmallattheIPF. A BFigure4-12. GasvelocityandPVFcontoursforthecaseA1=0:039145,A2=0:116199,A3=0:070466,k1=1,k2=12,k3=8,1=0,2=6:000171,3=0:378179.TheoutercontourrepresentstheinitialPV.Becauseofthechannelingeect,thegasmovesfasterinsectorswherethePVwasinitiallylower.A)Gasvelocitycontoursatnaltime,500s;B)PVFcontoursatnaltime,500s. InFig. 4-13 wepresenttheresultfortheleastampliedtrimodalcaseofA1=0:008722;A2=0:115284;A3=0:081446;k1=14;k2=2;k3=25;2=3:923277;3=3:135248.Againframe( 4-13A )showscontoursofgasvelocity,whileframe( 4-13B )showstheparticlelocationcoloredbyPVFbothatthenaltime(500s).TheinitialPVasafunctionofontheouterperipheryisplottedonbothframes.Thebiggestdierencebetweenthisandthepreviouscaseisthewavenumberofthemodethathasthelargest 96

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amplitude.InthecaseofFig. 4-12 thestrongestmodeofthetrimodalperturbationhasawavenumberofk=12,whilethelattercasethestrongestmodehasawavenumberofk=2.Thisdierencecontributestoasignicantlyalteredbehavior.InFig. 4-13 theperturbationisnotasstrongbothintermsofvariationinradialparticlepositionandintermsofPVvariation. A BFigure4-13. GasvelocityandPVFcontoursforthecaseA1=0:008722,A2=0:115284,A3=0:081446,k1=14,k2=2,k3=25,1=0,2=3:923277,3=3:135248.TheoutercontourrepresentstheinitialPV.Becauseofthechannelingeect,thegasmovesfasterinsectorswherethePVwasinitiallylower.A)Gasvelocitycontoursatnaltime,500s;B)PVFcontoursatnaltime,500s. 4.6.4ParticleLocationIntheprevioussection,wedescribedthechannelingeectwhichisresponsiblefor(i)increasingtheradialvelocityandtherefore,theradiallocationofparticlesinthelowPVsectorsand,(ii)movingparticlesfromsectorswithlowPVintoneighboringsectorsofhigherPV.Inthissectionwecomputeparticlevolumesweightedbypowersoftheirradiallocationandobserveditsdependenceonthecircumferentialdirection.Inparticular,weobservethatifwedividethedomainintocircularsectorsandwecalculateineach 97

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sectorthePV,butthistimeweightedbythelocationoftheparticles,weobtainthattheresultantproleisapproximatelyconstantintheazimuthalcoordinate,.Thatis,wefoundthat, NrXi=1PV(;ri)r2i=constant;(4-23)whereriisithcomputationalcellintheradialdirection,theazimuthalcoordinate,andNristhenumberofgridradialdivisionswhichinthiscase,are125.Eq.( 4-23 )canbeexplainedthefollowingway.ParticleslocatedinsectorswithinitiallylowerPVgainmoremomentum,whichincreasestheirspeed,achievingfurtherradiallocationsthanparticleslocatedinsectorswithhigherPV.However,wedonotknowhowmuchfastertheparticleslocatedinlowerPVsectorstravelcomparedwithotherslocatedinsectorswithhigherPV.Becausewewereinterestedingivingmoreimportancetoparticlesthattravelfurther,weweightedeachcomputationalcellPVwithitsradiallocationatthenthpower,rn,withn=1;2;3;4.ThusaradiallyfurtherparticlewillcontributemoretothePVsummation.Figure 4-14 showsthesimulationevolutionofPV()contoursalongwithrnPV()fortheunimodalcasewithk=8.Figure 4-15 showsthesameinformationthatFigure 4-14 butforthecase(A1;A2;A3;k1;k2;k3;2;3)=(0:13;0:04;0:05;8;17;15;2:05;4:84).Finally,Figure 4-16 showsthesameresultsbutforthecase(A1;A2;A3;k1;k2;k3;2;3)=(0:04;0:12;0:07;1;12;8;6:00;0:38).Intherstrow(PV)ofFigures 4-14 4-15 ,and 4-16 weseePVcontoursfordierenttimes.Thecontoursbecomelesswavyafterweighingwithr.whichcanbeseeninthesecondrow.WethenwentastepfurtherandweweightedthePVineachcellbythesquaredradiuslocation,r2.Weobservedthatthecontourslostallthewavinessexceptforsomenoisefortimeslaterthan200s,seerowthree(r2PV).Inotherwords,theintegralofPVtimesr2isapproximatelyequalinallthecircularsectors.Ofcourse,ifwekeepincreasingthepowerofr,theinitialmodalperturbationofthePVcontourwillbeippedduetothefactthattheradialweightistoolarge.Itisinterestingtoseeinthethirdrow 98

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(r3PV)andfourthrow(r4PV)thatattheendofthecompactionphase(t=100s)theinitialwavinessiscompletelylostandatlatertimesisrecoveredbutipped.Furtherresearchshowedthat N(j)pXi=1Vp)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(v(i)r2=constant;(4-24)whereN(j)pisthenumberofparticlesintheradialsectorjforj=1;2;:::;N,beingNthetotalnumberofradialsectorsconsidered,Vpisthevolumeofoneparticle,whichisoursimulationisconstantforalltheparticles,andv(i)ristheradialvelocityoftheparticleiinthesectorj. 99

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Figure4-14. EvolutionofthePV,rPV,r2PV,r3PV,andr4PVfortheunimodalcasek=8. 100

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Figure4-15. EvolutionofthePV,rPV,r2PV,r3PV,andr4PVforthetrimodalcase(A1;A2;A3;k1;k2;k3;2;3)=(0:13;0:04;0:05;8;17;15;2:05;4:84). 101

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Figure4-16. EvolutionofthePV,rPV,r2PV,r3PV,andr4PVforthetrimodalcase(A1;A2;A3;k1;k2;k3;2;3)=(0:04;0:12;0:07;1;12;8;6:00;0:38). 102

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4.6.5ShockandParticleBedEvolutionFinally,beforeinitiatingthecomparisonofinstabilityamongthemorethan1;800casesconsideredinthisstudy(Table 4-2 ),wewillstudytheeectofinitialperturbationsontheshockbehaviorandtheparticlebedevolution.Fig. 4-17 presentstheevolutionoftheshockwaveandtheparticlebedasafunctionoftime.Figure 4-17A showstheshocklocationasafunctionoftimeforthecases(i)withoutanyperturbationand(ii)withaunimodalperturbationwithk1=10.Ascanbeseen,theshockisnotaectedbytheperturbationsimposedinitiallyinthePVF.Thisbehavioristrueforalltheothercasesconsidered.Inallcases,theslowdecelerationoftheshockisthesameand,inallcases,theshockisnearlycylindricalanduninuencedbytheperturbation.Fig. 4-17B showstheazimuthallyaveragedvolumeofparticlesasthefunctionoftheradiusfordierenttimesfortheunperturbedcaseandthecorrespondingresultsforthetwotrimodalperturbedcasesthatshowedthelargestandthesmallestinstabilitygrowtharepresentedinFigs. 4-17C and 4-17D .Astobeexpected,theinitialvolumeofparticlesatt=0increaseslinearlywithradiussincethevolumeoftheannularregionsincreasesasr.Inallcases,themeanparticlelocationmovesradiallyoutastimegoes,andthelocationofthepeakscompare.Fortheunperturbedcase,thoughtheparticlebedseemstohaveradiallyexpanded,thebedappearstoremainrelativelycompact.ThisresultsinahighervolumefractionwiththemaximummeanPVvalue30%higherthanthosefortheperturbedcases.Intheperturbedcasestheradialextentofthebedislargerandasaresult,thepeakvolumeissomewhatlower.Thisbroadeningofthebedisclearlyduetolargerundulation(orwaviness)intheparticledistribution. 4.7Results:ComparisonofInstability 4.7.1TheNormalizedFourierEectiveVariationWeareinterestedinmeasuringtheamplicationofthedeparturefromaxisymmetrythatwasintroducedasaninitialperturbation.Thereareseveralwaystodenethisperturbation.Asaddressedintheprevioussection,theperturbationappearsbothas(i) 103

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A B C DFigure4-17. ShockandmeanPVlocationasafunctionoftheradius.Thelegendsrefertodierentsimulationtimes.A)Shockradiallocationvs.simulationtimeforthecasesconsidered,500s;B)AzimuthallyaveragedPVFvs.radiallocationfordierenttimesfortheunperturbedcase,500s;C)AzimuthallyaveragedPVvs.radiallocationfordierenttimesforthetrimodalcaseA1=0:039,A2=0:116,A3=0:070,k1=1,k2=12,k3=8,1=0,2=6:000,3=0:378;D)AzimuthallyaveragedPVvs.radiallocationfordierenttimesforthetrimodalcaseA1=0:009,A2=0:115,A3=0:081,k1=14,k2=2,k3=25,1=0,2=3:923,3=3:135. 104

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-variationintotalPVand(ii)-variationintheradiallocationoftheparticles.SincetheperturbationwasintroducedintermsofvariationinPV,herewewillfocusonquantifyingthegrowthinthisperturbationbyintroducingthenormalizedeectivePVvariation,F,asametricofgrowthquantication.Wedividethecomputationaldomainintoradialsectorsofidenticalvolume.Inourproblem,thevolumeofthesectorsremainsaconstantandequaltohR2=N,whereN=256isthenumberofgridcellsinthecircumferentialdirection,R=0:6mischosentobetheouterradiusofthecomputationaldomain,andh=0.02misthethicknessofthecomputationaldomainintheaxialdirection.Notethateventhoughthegaspropertiesaretwo-dimensionalanddonotvaryalongtheaxialdirection,particlesaredistributedwithinthe3Ddomainoverthisaxialthickness.Foranysimulationtimet,wecalculatethetotalvolumeofalltheparticleswithineachofthe256radialsectors,whichdenesthevariablePV(;t).Becauseofthecylindricalnatureofthephysicalproblem,PVisaperiodicfunctionin02.WecanFouriertransformPV(;t)asfollows PV(;t)=N=2Xk=)]TJ /F6 7.97 Tf 6.58 0 Td[(N=2akexpik2 N;(4-25)whereakistheFouriercoecientcorrespondingtothekthFouriermode.TheFouriercoecientsarecomplexandaregivenby ak=1 NN)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xj=0PV(j;t)exp)]TJ /F4 11.955 Tf 9.3 0 Td[(ik2 Nj:(4-26)Aplotofjakj2asafunctionofthewavenumberkgivesustheenergyspectrumofdeparturefromaxisymmetryintheperiodicsignalPV(;t).Figs. 4-18A and 4-18B showthespectrumofthemostampliedtrimodalcaseattheinitialtimeandatt=500s,respectively.Thespectraarenormalizedbythesquaredaveragevalueja0j2andtheresultsaresymmetricinthewavenumber. 105

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A BFigure4-18. PVamplitudespectrumsquaredforA1=0:130;A2=0:039;A3=0:039,k1=8;k2=17;k3=15,and1=0:200;2=0:520;3=4:851.ThespectrumisnormalizedbyjA0j2.A)PVamplitudespectrumsquaredattheinitialtimet=0;B)PVamplitudespectrumsquaredatthenaltimet=500s. InFig. 4-18A ,besidesthantheconstantmeanoftheperturbation(i.e.k=0),onlytheinitialmodes[k1;k2;k3]=[8;17;15]thatareimposedbytheinitialperturbationhaveanon-zeroamplitude,whiletherestofthemodesarezero.Atthelatertimeoft=500sweobservetheinitialthreemodestoremainstilldominantandtheirsquaredamplitudeshavesubstantiallygrownovertime(notethelogarithmicy-axis).However,alltheotherFouriermodesareenergizedaswell.ThisispartlyduetononlinearinteractionbetweentheinitialFouriermodes,butalsoduetocircumferentialperturbationintroducedbytherandomlocationandmotionoftheparticles.ThetimeevolutionofFourieramplitudesforatypicaltrimodalperturbationisshowninFig. 4-19 .Frame( 4-19A )showstheevolutionofthethreemodesk=7;15and25thatwereinitiallyperturbed.Allthreemodesrapidlygrowinamplitudeovertherst200microseconds,afterwhichmodesk=7and15continuetogrowataslowerratewhilemodek=25showsaslowdecayafterreachingapeakataroundt=250s.Frame( 4-19B )showsthetimeevolutionoffourothercircumferentialmodesthatwerenotinitiallyperturbed.Thesemodescanbeexpressedaslinearcombinationsoftheinitially 106

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perturbedmodesas:(i)k=10=(25)]TJ /F1 11.955 Tf 11.97 0 Td[(10),(ii)k=18=(25)]TJ /F1 11.955 Tf 11.97 0 Td[(7),(iii)k=54=3(25)]TJ /F1 11.955 Tf 11.97 0 Td[(7)andk=72=4(25)]TJ /F1 11.955 Tf 12.03 0 Td[(7).Itcanbeseenthatthemodesk=10and18growovertheentiredurationofsimulation,perhapsduetothenonlinearinteractionbetweentheinitiallyperturbedmodes.Incontrast,thehighermodesatthebeginninggrowevenfaster.Theirinitialperturbationisfromtherandomdistributionofparticlesandnotfromthenonlinearmodalinteraction.Therapidgrowthlastsonlyforashorttime,followedbyadecayandfort>200sasecondslowergrowthphasecanbeobserved. A BFigure4-19. Amplitudeevolutionofinitialandemergingmodes.A)Amplitudeevolutionforinitialwavenumbers7,15and25.At500stheamplitudeisroughlyampliedbyafactorof2.k=7andk=15arestillgrowingwhilek=25seemstobedecaying;B)Amplitudeevolutionforemergingwavenumbers10(25-10),18(25-7),54(3(25-7))and72(4(25-7)).At500stheamplitudesreachroughly25%oftheamplitudeofinitialmodes.k=18andk=72amplitudesdoubletheothers. WedenethenormalizedeectivePVvariationas F(t)=PN)]TJ /F5 7.97 Tf 6.59 0 Td[(1k=1a2k A21+A22+A23:(4-27)Notethatinthenumeratorthesumexcludesk=0andthusonlythenon-axisymmetricmodescontributetothismeasure.Also,thedenominatoristhesumofsquaredamplitudesofthethreemodesoftheinitialperturbation,whichinthepresentsimulationshas 107

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beenchosentobeequalto0:02.ThedenominatorisnothingbuttheinitialvalueofthenumeratorforthepresentsetofsimulationsandthereforewehavenormalizedtheabovemeasureofPVvariationtoyieldaninitialvalueofF(t=0)=1.Forexample,forthecasepresentedinFig. 4-18B at500sthevalueofFis6.25. 4.7.2TimeEvolutionandStatisticalNoiseFigure 4-20 showsthetimeevolutionofthemetricFforselectedcasesoftrimodalperturbation.Itwillbeshownlaterthatamongthemanyparametersoftheproblem,thecriticalonewhichcontrolsthegrowthoftheperturbationistheeectivewavenumberdenedin( 4-22 ).Ingeneral,largereectivewavenumbersgenerateastrongerperturbationgrowth.However,theotherparametersoftheproblemalsoplayaroleandcontroltheoverallgrowthrate.Thelargenumberoftrimodalsimulationsaregroupedintosetsaccordingtotheirvalueofkeff.IneachoftheframesofFig. 4-20 ,theresultsofonlythosecaseswhosekeff=5;10;15and20areshown.Inallcases,Fincreaseswithtimeandtherateofincreaseislargerforlargereectivewavenumber.Despitethisgeneralincreasingtrend,thereissubstantialvariationbetweenthedierentcasesillustratingtheroleofparametersotherthankeff.Inthekeff=5cases,themetricshowsamodestincreasetoabout2to4att=500s.Inthekeff=10casesFhasincreasedtofromabout4to6.5,whileinthelargertwoeectivewavenumbercasesthenalvalueofFisintherange6to9.Thedierencebetweenthecasesofkeff=15and20isnotlarge.Thisdependenceofgrowthrateonkeffwillbeexploredfurtherinthelatersections.Inallthecases,wecanobserveaninectionpointinthebehaviorofF(t).Inotherwords,attheverybeginning,Fincreasesveryrapidlywithtime.Thisrateofincreaseslowsdownovertimeasindicatedbytheinectionpoint.Thetimecorrespondingtothisinectionpointislowerforsmallerkeff.Thus,forkeff=5,therapidlyincreasinggrowthrateislimitedtoashortperiod,followedbyalongperiodofeverslowinggrowthrate.Whereasforkeff=10and15theinectionpointisaroundt=250s,tillthenthegrowthrateisincreasingandtheslow-downofgrowthisonlyafter. 108

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A B C DFigure4-20. ThemetricFasafunctionoftimeforfourdierentkeff.Eachlinerepresentadierentcase(i.e.dierentA1;A2;A3;k1;k2;k3;1;2;3butwithroughlythesamekeff).BydenitionthemetricFisoneatt=0,therangeofFatt=500sstronglydependsonthekeffvalue.A)ThemetricFasafunctionoftimeforperturbedcaseswithkeff=5;B)ThemetricFasafunctionoftimeforperturbedcaseswithkeff=10;C)ThemetricFasafunctionoftimeforperturbedcaseswithkeff=15;D)ThemetricFasafunctionoftimeforperturbedcaseswithkeff=20. 109

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Figure 4-20 clearlysuggestthepresenceofsubstantialstatisticaluctuationsinF.PartofthevariationinFcanbeexplainedintermsofparametricdependence,butthecomplexoverlappingbehaviorseeninFig. 4-20 cannotbefullyexplainedintermsofthecontrollingparametersalone.Oneofthecontributingfactorstothestatisticalnoiseistheinitialrandomdistributionofcomputationalparticleswithintheannularregion.Thepreciseparticlepositionsineachsimulationweredecidedbasedonrandomnumbergenerators.Thus,theactualpositionofparticlesvariesfromcasetocase.Eventhesamecasewithidenticalmodalperturbationparameterswillyieldastatisticallydierentresultiftheinitialdistributionofparticlesisvariedusingadierentrandomseed.Forthesameinputparameters,thedierentsimulationswithvaryingparticledistributionscanbeconsideredasstatisticalrealizations.ThevariationinFamongthesestatisticalrealizationscanbeconsideredasthestatisticalnoise.Astheparametersoftheproblemarevaried,theimportanceoftheseparametersbecomerelevantonlyiftheireectonFissubstantiallylargerthanthestatisticalnoise.Figure 4-21 showsingrayaboxplotbuiltusingthe1,600trimodalperturbationsimulationresultsmentionedinTable 4-2 .ThevalueofFatt=500sfromallthesimulationsareusedtoconstructtheboxplot.Thesevenparametersofthetrimodalperturbation(A1;A2;k1;k2;k3;2;3)wererandomlyselectedusingLatinhypercubesamplingtechnique(recallthatthephaseoftherstmode1canbesettozerowithoutlossofgeneralityandA23=0:02)]TJ /F4 11.955 Tf 12.37 0 Td[(A21)]TJ /F4 11.955 Tf 12.38 0 Td[(A22).ThemetricFhasameanof6.3,amedianof6.9andarangeof[1.2,8.7].Thus,astheparametersofthetrimodalperturbationarevariedwhilemaintainingtheoverallstrengthoftheinitialperturbation(i.e.,A21+A22+A23=0:02),thegrowthoftheperturbationcanvaryalmostanorderofmagnitude.Toexaminethestatisticalsignicanceoftheobservation,wewillnowestablishthelevelofstatisticalnoiseduetotherandominitiallocationofparticles.Threeorangeboxplotsarepresented,eachonerepresentingonecasewhereallthevariablesofthetrimodalperturbationarexed.OneisconstructedusingacasewithF(t=500)=1,oneforF(t=500)=5, 110

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andoneforF(t=500)=9.Thevariabilityintheseplotsisduetothedierentrandomdistributionofparticles(usingdierentrandomseedsinthegenerationoftheinitialcomputationalparticlelocationswithinthecomputationaldomain).Eachoftheseboxplotsisbasedon20dierentstatisticalrealizationsofthesamecase.Notethatwhentheseedischangedonlythelocationofthecomputationalparticleschanges,whilethePVFinthecelldoesnotchange.TherelativenoisewasestimatedastwotimesthestandarddeviationinFvaluesofthe20realizationsnormalizedbythemeanvalueofthe20realizations.ThenoiseestimationfortheF(t=500)=1,F(t=500)=5,andF(t=500)=9boxplotsare3.7%,4.8%,and4.1%,respectively.Conservatively,wecansaythatthenoiseinherenttoourproblemis5%forthemetricF(t=500s).Havinganestimationofnoiseisveryimportantfortworeasons.Therstoneisthatitallowsustoestablishtheextenttowhichwecantrustthemetricvalueobtainedinanysimulation,i.e.nowweknowthatwehaveanuncertaintyofabout5%.Thesecondreasonisthatittellsusthatwhencomparingtheresultsoftwodierentcasesofdierentinitialperturbations,theirdierencemustbesubstantiallylargerthanthestatisticalnoisetobeconsideredsignicant. 4.7.3TheWeakDependenceonPhaseIntheprevioussection,itwasdeterminedthatthemetricFhasarelativenoiseofabout5%duetotherandomnessinthewaythatparticlesareinitiallylocatedwithinthedomain.WenowinvestigatethesensitivityofFtoeachoneoftheparametersinvolved.Wedeterminedthattheleastsensitiveparameterswherethemoderelativephases.Itisimportanttoknowiftheinuenceofavariableinthemetricishigherthanthenoise.Therefore,Figure 4-21 showsthreeadditionalboxplotscoloredblue.AgaineachofthenewboxplotsisassociatedwithacasewithF=1,F=5,andF=9byvarying2and3randomlyintheinterval[0;2]andeachofthesenewboxplotswasbuiltusingalso20simulations. 111

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Figure4-21. Variabilityinthemetric.TheleftmostboxplotshowstheentirerangeofF(t=500s).OrangeboxplotsshowsF(t=500s)variabilityforacasesusingthesameparametersbutwith20dierentrandomseedsinthelocationofparticleswithinacell.Blueboxplotsshowthevariabilityofcasesusingrandomlyselectedphasesfor20simulationseach. WedeterminedthatthevariabilityofFduetorandomvaluesofthephasesbetweenthemodesoftheinitialPVFare5.8%,4.9%,and5.1%forthethreeboxplots.Conservativelywecanstatethatwehaveupto6%variabilityinthemetricFduetophasevariation.Therefore,weobservethatnotonlydo1;2;3havethelowestinuenceonthemetric,moreover,theirinuenceiscomparablewiththerandomnoiseinthesimulations. 4.7.4TheStrongDependenceontheEectiveWavenumberInthissection,weshowhowthemetricFvarieswithkeffaidedbytheresultsof1,725simulations.Figure 4-22A showsF(t=500s)asafunctionofkefffortheunimodalinitialperturbations(green),bimodalinitialperturbations(orange),andfortrimodalinitialperturbations(blue).Thegurewasbuiltusingdatafromsimulationsof25unimodalinitialperturbations,100bimodalinitialperturbations,and1,600trimodalinitialperturbations. 112

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Firstandforemost,althoughnotaperfectcollapse,keffseemstocollapsethedependenceonthethreewavenumberparametersintoasinglevariable.Otherdenitionssuchas(k1+k2+k3)=3orp (k21+k22+k23)=3donotcollapsethedataaswell.Theweightingnormalizedamplitudeinthedenitionofkeffisimportantingivingappropriateweighttothedierentmodes.Thelackofperfectcollapseisduetoboththedependenceonthephaseand,moreimportantly,onthedistributionofmodalamplitudes.WeobservethatFappearstogrowlinearlyforkeff<10,afterwardsthereisaplateaufor10keff20,andnallyFdecreasesforkeff>20.Themetricseemstobemaximizedbyunimodalperturbations.Thatis,imposingaunimodalperturbationwillgivethemaximumFatnaltime(500s).BimodalperturbationsleadtovaluesofFintheentirerangeofthetrimodalperturbations.StudyinghowFdependsonkeffisawaytostudyhowFdependsonthewavenumber.Figure 4-22A showsastrongdependenceofFonthewavenumber.ThehighestvalueforFwasfoundusingasinglemodelperturbationwithkeff=20. A BFigure4-22. F(t=500s)asafunctionofkeff.A)F(t=500s)asafunctionofkeffforunimodal,bimodalandtrimodalinitialperturbations;B)F(t=500s)asafunctionofkeff.Thecolorbarrepresentsthehighestmodeamplitudeoftheinitialperturbationsquaredandnormalized. 113

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4.7.5TheDependenceontheModeAmplitudeHereweconsiderthedependenceofthemetricFontheinitialmodalamplitudes.ArstapproachishelpedbyFigure 4-22B ,whichshowsthetrimodalcasesofFigure 4-22A coloredaccordingtotheamplitudeofthemodewiththehighestamplitudenormalizedbytheconstantA21+A22+A23=0:02.Notethat,byconstruction,thecolorbarrangegoesfrom1/3,correspondingtothecaseofequipartitioningofenergybetweenthethreemodes(i.e.A1=A2=A3),to1,correspondingtotheunimodallimit.Forkeff<5,F(t=500s)hasthehighestvaluesforcasesroughlywithA1=A2=A3,howeverforkeff>5,F(t=500s)seemstobemaximizedbyasinglemodeperturbation.Figure 4-23 showsthecasesshowninFig. 4-22B butwiththedatadividedintodierentbinsofkeff.Hereeachaxisisassociatedwithaninitialamplitude,thereforethepointslieonthesurfaceofthesphereofradiusp 0:02.ThecoloringrepresentsthevalueofF(t=500s).AswecanseeinFigure 4-23A ,forlowkeffweobtainahigherF(t=500s)frommultimodalinitialperturbations.Ontheotherhand,for520(Figure 4-23E )wedonotseeasclearapatternasinthepreviouscases,butF(t=500s)seemsstillbemaximizedinthecorners.ThisbehaviorcanbeobservedalsoinFigure 4-22A whereunimodalperturbationsareshowingthehighestmetricvalues.ItisalsoseeninFigure 4-22B wheretheunimodalperturbations(max(A21;A22;A23)=0:02=1)showsthehighestmetricvaluesforkeff>5. 4.7.6TheNormalizedMaximumParticleVolumeFractionDierenceTheconclusionsoftheprevioussectionclearlydependonthedenitionofthemetricFbeinginvestigated.Thesensitivityoftheresultswasstudiedbyconstructingseveralothermeasuresofinstabilitygrowth.Inthissection,wedescribeanothermetricthatwerefertoastheNormalizedMaximumParticleVolumeDierence,.Thismetricalso 114

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A B C D EFigure4-23. Positiveoctantoftheamplitudespherewithradiusp 0:02fordierentrangesofkeff.ThecolorbarshowsthemetricF(t=500s)inthatinterval.A)0
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measurestheamplicationofthedepartureofinitialaxisymmetry.WhileFhasaglobalnature,isbasedononlyextremes.Tocompute,thePV(;t)iscalculatedasexplainedinEq.( 4-25 )andiscalculatedasafunctionoftimeas (t)=PV(t) PV(t=0);(4-28)wherePViscomputedas PV(t)=max(PV(;t)))]TJ /F1 11.955 Tf 11.95 0 Td[(min(PV(;t));(4-29)wheremax(PV(;t))isthePVofthesectorwithmostparticlesattimet,andmin(PV(;t))isthePVofthesectorwithleastparticlesattimet.Attheinitialtime,,asF,is1byconstructionforallcases.AreproductionofFig. 4-22A ,butforthemetric,showedasimilargeneralbehaviorasF,however,thegrowthasafunctionofkeffisslowerandthedispersionofthedataishigher.Inthiscase,aswell,sensitivityanalysesalsoconcludedthatthephaseparametershaveanegligibleinuenceincomparisonwiththewavenumberandamplitudevariables.Usingboxplotswefoundthathasa10%ofnoiseduetorandomnessintheinitiallocationofparticleswithinthedomain.ThehighernoiselevelistobeexpectedsinceFcanbeconsideredasanL2metricwhileisanL1metricthatdependsonpointwisevalues.Thereforethephaseparameteris,forthismetric,evenlessinuential.Wealsofoundthat,likeF,ismaximizedbyunimodalinitialperturbationsandthatbimodalperturbationsleadtoresultsthroughouttherangeof.Thehighestvalueforwasfoundusingaunimodalperturbationwithk1=21.AsinFig. 4-22B thehighestpossibleamplitudesseemtohavethehighest(unimodalcases)forkeff>5.Thisreinforcesthefactthatsinglemodesmaximizethedeparturefromaxisymmetryatnaltime.Weconclude,therefore,thatbyusingtwodierentmetrics,Fandweobtainthesamebehavior. 116

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4.8ConcludingRemarksInthissection,weinvestigatedtheevolutionofsmallperturbationsintheinitialparticlevolumefractionofparticledistributionwithinanannularbedsurroundingahigh-energyexplosiveinacylindricaltwo-dimensionalmultiphaseexplosion.Theinitialperturbationsweresinusoidalintheangularcoordinateandconstantintheradialcoordinate.Weusedcombinationsofuptothreemodeswhilevaryingtheirparameters(amplitude,wavenumber,andrelativephase).Weobservethattheradialsectorswithlargerparticlevolumefractionsappearedasporousnozzleswithalargerareacontraction,whilesectorswithsmallerparticlevolumefractionsappearedasporousnozzleswithasmallerareacontraction.Thepost-shockowinthepresentproblemwassupersonic,thereforethenozzlingeectwastodecreasethevelocityinregionsoflargervolumefraction.Thisreductionwasfurtherstrengthenedbytheincreaseddragonthegasowduetothehigherconcentrationofparticlesinthehighvolumefractionsectors.Theparticleslocatedinthehigh-speedsectorsalsogainedmoremomentumthanparticleslocatedinradialsectorswithhigherparticlevolumefraction.Astheparticlesinthelowvolumefractionsectorsmovedfartheroutradially,theyalsotendedtocircumferentiallymigrateintotheslowmovinghighvolumefractionsectors.Thus,sectorsofhighvolumefractiontendedtofurtherincreaseattheexpenseoflowvolumefractionsectors.Thisfeedbackmechanismwasthesourceofthechannelinginstabilityobservedinthepresentsimulations.Theneteectofthechannelinginstabilitywastoincreasestheangularvariationinthenetvolumeofparticlescontainedwithinradialsectorsofthedomainandalsotoincreasethedierenceintheradialextentofparticleswithinthedierentsectors.Boththesemechanismsappearedasdeparturesfromaxisymmetry.Wesoughttomeasuretheamplicationoftheseinstabilitiesandthedeparturefromaxisymmetryintheparticledistribution. 117

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Wedenedtwometrics,Fand,inordertoobtaininformationregardinghowtheparticlesaredistributedatlatertimesinthesimulations.Fisametricthatdependsonthetotalamplitudeofallthecircumferentialmodespresent,includingboththeoriginalperturbationmodesandtheemergingnewones.isametricthatdependsonlyonthedierencebetweenradialsectorsofmaximumandminimumvolumeofparticles.Weobservethattherelativephasesbetweenthemodesoftheinitialperturbationdonotplayanimportantroleinthemetricswhencomparedwiththeotherparameters(amplitudesandwavenumbers)and,furthermore,theirinuenceinthemetricsisontheorderofthenoise.Wealsondthattheeectivewavenumber,avariableconstructedasthesumofthewavenumbersweightedwiththesquaredamplitudes,tohavethestrongestinuenceonthemetrics,morethananyothervariable.WeconcludethatbothmetricsFandaremaximizedatnaltime(500s)byinitialunimodalperturbations.Thisledtotheconclusionthatthenetsquaredamplitudesorthemaximumdierencebetweenthenumberofparticlesinaradialsector,ishighestifweusedunimodalperturbationsratherthanbimodalortrimodalperturbations.TheunimodalthatmostampliedthemetricsFandarek1=20andk1=21,respectively.Butthewavenumberofthispeakmodecanbeexpectedtobedependentonthecircumferentialresolution. 118

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CHAPTER5ONTHEUSEOFSYMMETRIESINBUILDINGSURROGATEMODELS 5.1SummaryWhensimulationsareexpensiveandmultiplerealizationsarenecessary,asisthecaseinuncertaintypropagation,statisticalinference,andoptimization,surrogatemodelscanachieveaccuratepredictionsatlowcomputationalcost.Inthiswork,weexploreoptionsforimprovingtheaccuracyofasurrogateifthemodeledphenomenonpresentssymmetries.Thesesymmetriesallowustoobtainfreeinformationand,therefore,thepossibilityofmoreaccuratepredictions.Ananalyticalexamplealongwithaphysicalexamplethathaveparametricsymmetriesarepresented.Althoughimposingparametricsymmetriesinsurrogatemodelsseemstobeatrivialmatter,thereisnotasinglewaytodoitand,furthermore,theachievedaccuracymightvary.Fourdierentwaysofusingsymmetryinsurrogatemodelsarepresented.Threeofthemarestraightforward,butthefourthisoriginalandbasedonoptimizationofthesubsetofpointsused.Theperformanceoftheoptionswascomparedwith100randomdesignofexperimentswheresymmetrieswerenotimposed.Wefoundthateachoftheoptionstoincludesymmetriesperformedthebestinoneormoreofthestudiedcasesand,inallcases,theerrorsobtainedimposingsymmetriesweresubstantiallysmallerthantheworstcasesamongthe100.Theoptionsforusingsymmetriesintwosurrogatesthatpresentdierentchallengesandopportunitieswereexplored:Krigingandlinearregression.Krigingisoftenusedasablackbox,thereforeweconsiderapproachestoincludethesymmetrieswithoutchangesinthemaincode.Ontheotherhand,sincelinearregressionisoftenbuiltbytheuser,owingtoitssimplicity,weconsideralsoapproachesthatmodifythelinearregressionbasisfunctionstoimposethesymmetries. 5.2Nomenclature p:Particlevolumefraction(PVF) 119

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p0:BasePVF Ai:Amplitudeofthemodei ki:Wavenumberofthemodei i:Phaseofthemodei :Maximumnumberofparticlesinacircularsector :Normalizedparticlevolumedierence ^y:Approximationofyusingnutrainingpoints m:Numberoftestpoints ^yAPP:(Addpermutationpoints)Approximationofthefunctionyincludingthepermutationpointsastrainingpoints ^yRD:(Restricteddomain)Approximationofthefunctionywithsamplesrestrictedtothedomainx1
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Nt:Setofntdatapoints(Nt=No[Ns) Nu:Setofnudatapointsusedtobuildthesurrogatemodel(NuNt) 5.3BackgroundDespitegreatstridesincomputationalpoweroverthepastfewdecades,investigationofcomplexproblemsthroughcomputationalsimulationsremainsachallenge.Therefore,whensimulationsarecomputationallyexpensiveandmultiplerealizationsareneeded,asinuncertaintyquantication(UQ)[ 8 { 10 ],inverseproblems[ 11 { 13 ]oroptimization[ 14 { 16 ],surrogatemodelsbecomeanattractiveoption.Mostsurrogatesarealgebraicmodelsthatapproximatetheresponseofasystembasedonttingalimitedsetofcomputationallyexpensivesimulationsinordertopredictaquantityofinterest.Somesurrogatescombinemultiplemodeldelities[ 17 18 ].Theiraccuracyisinuencedbythedesignofexperiments(DoE)used,thesizeofthedomainofinterest,thesimulationaccuracyatthedatapointsandthenumberofsamplesavailablefortheirconstruction[ 19 ].Therearetwopopularsurrogatemodelsthatwewilluseinthiswork:linearregressionandKriging.Responsesurfacemodels[ 262 ],suchaslinearregressionmodels,aretheoldestsurrogatesandtheymaystillbethemostwidelyusedinengineeringdesign.Thesearechosenduetotheirsimplicityandlowcostoftting.Theyonlyrequirethesolutionofasystemoflinearalgebraicequations.Responsesurfacemodelsgenerallyassumethatthefunctionalbehavioriscorrectbutthedatapointshavenoise.Ontheotherhand,Krigingestimatesthevalueofafunctionasthesumofatrendfunction,representinglow-frequencyvariation,andasystematicdeparture,representinghigh-frequencyvariationcomponents[ 82 ].Unlikeresponsesurfacesmodels,Krigingusuallyassumesthatthedataiscorrectbutthefunctionalbehaviorisuncertain.Symmetrieshaveplayedanimportantroleinthemodelingofcomplexprocesses.Forexample,Beatovicetal.,1992[ 263 ]proposeaGalerkinformulationformodelingaxisymmetricproblemsinElectrostatics.Rahrahnamayanetal.,2008[ 264 ]useopposites 121

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datapointstoaccelerateconvergenceindierentialevolutionoptimization.Baietal.,2009[ 265 ]reducetheoptimizationdomaineliminatingredundantinformationduetosymmetries.Dengetal.,2015[ 266 ]presentaninitialpopulationstrategytoimprovethegeneticalgorithmforsolvingasymmetrictravelingsalesmanproblem.Symmetriesinherenttoaproblemmightleadtocostsavingsinsurrogateconstruction.Mathematically,manydierentsymmetriescanbedened.Themostcommonaresymmetriesbasedongeometry,suchasreectionalsymmetry,rotationalsymmetry,translationalsymmetryandsoon.Forexample,inthecaseofalaminarowthroughasquareduct,thesolutionmustsatisfyreectionalsymmetryaboutthemidplanesandaboutthediagonals,andsatisfyrotationalsymmetryto90,180and270rotations[ 267 ]asshownschematicallyinFigure 5-1 .Otherexamplesarethecyclicsymmetryofcascadeblades[ 268 ],thereectionalsymmetryofelectricandmagneticdipoles[ 269 ],andtrussstructureswithrotationalsymmetry[ 265 ]. Figure5-1. Schematicofthevelocityeldofalaminarowinasquareduct.Itsatisesreectionalsymmetryaboutthemidplanesandaboutthediagonals,andsatisesrotationalsymmetryto90,180,and270rotations Thesesymmetriescanalsobeconsideredintheparametricspace.Forexample,thefunctiony(x1;x2)oftheanalyticalexampleincludedinthispaperhasreectional 122

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symmetrywithrespecttothex1=x2axisthatcanbealsoviewedasparametricsymmetry,i.e.y(x1;x2)=y(x2;x1).Inthephysicalexampleconsideredinthiswork,k1,k2andk3arethewavenumbersofthethreedisturbancemodesthatareintroducedintheinitialconditions.Inthiscase,theorderoftheparametershasnosignicance.Thatis,theparameters(wavenumbers)couldbeorderedas(k1;k2;k3),or(k3;k1;k2),or(k2;k1;k3),etc.Suchparametricsymmetryisknownaspermutationsymmetry.Herewewillstudyhowtheparametricsymmetriesofaproblemcanbeexploitedintheconstructionofitssurrogates,analyzingtheperformanceofthesurrogateassociatedwiththedierentoptionsofimposingsymmetries.Summarizing,thispaperpresentsfourdierentalternativestobuildsurrogatemodelsimposingthesymmetriesalreadypresentinaproblem.Itwasfoundthat(i)therearesubstantialdierencesinperformancebetweenthedierentoptions,(ii)thereisnotoneoptionthatoutperformsothersconsistentlyinallcases,and(iii)thereiseventhepotentialofincreasingtheerrorbytakingadvantageofsymmetry.AdescriptionoftheproposedalternativescanbefoundinSection2.Threearestraightforwardoptionsalreadyusedbyresearcherstotakeadvantageofsymmetries,oneisnotasstraightforwardandwewouldliketohighlightitasoneofthecontributionsofthispaper.Abriefintroductionofthesurrogatemodelsused,Krigingandlinearregression,isincludedinSection 5.5 .TheanalyticalexampleispresentedinSection 5.6 ,whilethephysicalexampleisincludedinSection 5.7 5.4ApplyingSymmetriesLetusconsideradvariablefunctionf(x),wherex=(x1;x2;x3;:::;xd),suchthattheorderofthevariablesdoesnotmatter,i.e.f(x1;x2;x3;:::;xd)=f(x2;x1;x3:::;xd)=f(x3;x1;x2:::;xd)andsoon.Therefore,thefunctiontakesthesamevalueatd!points(thedatapointanditspermutations).Thesepermutationsoftheoriginaldatapointwillbecalledpermutationspoints. 123

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Ifweallowrepetitionsofthevariablevaluesinthesamedatapoint,thenumberofpermutationpointswouldbeless.Forexample,ifthevalueofvariablex1andvariablex2arethesame,thenthefunctionwilltakethesamevalueatd!=2!points.Thisnumberwilldecreaseevenmoreifthenumberofrepetitionsincreases.Inthispaper,wepresentananalyticalproblemthatallowsrepetitions(i.e.pointsinthelineofsymmetry),andaphysicalexamplethatdoesnot.LetNobethesetoforiginaldatapointswithnoelements.LetNsbethesetoffreelyavailable,duetosymmetry,nsdatapoints.LetNtbethesetoftotaldatapoints(Nt=No[Ns).Finally,letNubethesetofnudatapointsusedtobuildasurrogatemodel,whereNuNtandnunt.Inthispaper,wedonotallowpermutationpointsasoriginaldatapoints,e.g.if(x1;x2;x3;:::;xd)2Nothenthepermutationpoint(x2;x1;x3;:::;xd)=2No.Then,westudyandcomparetheaccuracyofapproximationsconstructedusingNutrainingpointsinfourdierentways.Threeofthemarestraightforwardanddonotneedsubstantialexplanation;however,thefourthisoriginalandrequiresamoredetaileddescription. TheStraightforwardOptions:APP,RD,andSBF { Addpermutationpoints(APP):BuildingthesurrogateusingNu=Ntastrainingpoints. { Restricteddomain(RD):InthiscaseNu=fx2Ntjx1x2x3xdg.Notethatascendingordescendingorderdoesnotmakeanydierence.Thiswaythedatapointsareclosertooneanotherallowingabetterapproximationinthatregion.Extensionbeyondthatregionwillbecalculatedbyreectingthesolutionassymmetriesdictate.Thisachievesbothobjectivesofasmallernumberoftrainingpointsandcloselypackedtrainingpoints.However,likeAPP,itmaysuermorefromttingnoise. { Symmetricbasisfunctions(SBF):Herewemodifythebasisfunctionsofthesurrogate.Duetothesymmetries,thenumberofunknownsurrogatecoecientsisreduced,hencewearelikelytoobtainamoreaccurateandcomputationallyfastersurrogatemodel.InthiscaseNu=No.Themodicationofthebasisfunctionissurrogatedependent,weexplainindetailtheprocedureforKrigingandlinearregressionsurrogatesinSection 5.5 124

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TheNovelOption:OSOptimalset(OS):Surrogatesbecomeexpensiveandnumericallyill-conditionedwithalargenumberofdatapoints.Inthiscase,wecanuseasNudierentsubsetsofNtwithnodatapoints,thatiswithoutincreasingthenumberofpoints.TheDoEthatminimizesthefreerootmeansquareerror(freeRMSE)maybechosen.ThefreeRMSEistherootmeansquareerror(RMSE)thatdoesnotneedextrasimulationsanditiscomputedusingastestpointstheremainingNt)]TJ /F4 11.955 Tf 12.77 0 Td[(Nudatapoints.Itminimizesameasureofthedeparturefromsymmetry.NotethatthefreeRMSEisjustanapproximationtotheRMSEcalculatedinthisworkatadditionaltestpoints,therefore,unliketheRMSEitcomesatnoextracost.Inthiswork,wechosetheOSfrom100DoEs.AddingmoreDoEs(wecomputedupto200)doesnotchangethefreeRMSEsubstantially(lessthan2%).However,thenumberofDoEsneededwillbeproblemdependent.EachDoEisgeneratedbyaddingonerandompointatatimeandndingitsnearestneighborinthesetNt.ThisisrepeateduntilN0pointsaregenerated.NotethatOSdoesnotresultinaperfectlysymmetricsurrogate.MinimizingthefreeRMSEleadstothemostsymmetricsurrogatethatcanbegeneratedwithnodatapoints.Figure 5-2 isasimpleexampleofthesetNuthatAPP,OS,andRDapproacheswouldgiveasatrainingDoE.Inthiscase,Nohasno=3datapointsshownintheleftsubgureusingemptycircles.ThesubgurealsoshowsthecorrespondingNt)]TJ /F4 11.955 Tf 12.67 0 Td[(Nopermutationpointsusingcrosses.ThelledcirclesintherightsubguresrepresentNuforeachoftheapproaches,OS,RDandAPP.TheupperrightsubgureistheDoEfortheAPPapproachwhereNu=Nt.ThemiddlerightsubgureshowsthehypotheticalDoEforwhichthefreeerrorestimateisthelowestamongthetestedones.ThiscorrespondstoOSapproachwhichusesasDoE,inthiscase,oneoftheNopointsandtwooftheNsones.Thefreeerrorestimatecanalsobecalculatedusingcross-validationerror,butwehaveobtainedbetterresultsincomputingtheerrorwiththeremainingNt)]TJ /F4 11.955 Tf 12.4 0 Td[(Nupermutationpoints.WedenotethiserrorestimateasfreeRMSE.Finally,thelowermostrightsubgureistheDoEforRDwhereNu=f(x1;x2)jx1
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Figure5-2. ExampleofaDoEforAPP,OS,andRDapproachesfortheuseofsymmetries.EmptycirclesaretheoriginalDoEofno=3points(N0),crossestheircorrespondentns=3permutationpoints(Ns).Fullcirclesintherightsubguresaretheresultingnudatapointsforeachapproach(Nu) regressionusersusuallyworkwiththeirownscriptand,unlikeKriging,isnotusuallyablackboxsurrogate.Consideringthis,wedomodifythelinearregressionbasisfunctionsimposingthesymmetriespresentinourproblem.Therefore,forlinearregression,weexplorealltheoptionsdescribedinthepreviouslist:APP,SBF,RD,andOS.Kriging,althoughmorecomplex,generallyismoreaccuratethanothersurrogatesifthedataisnotnoisy.Krigingisoftenusedasablack-boxsurrogate,wherethebasisfunctionscannotbemodied.Takingthisintoconsideration,SBFforKrigingwillbeoutofthescopeofthispaperbutAPP,OS,andRDwillbetested. 126

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Now,wewilldenetheerrorpredictionmetricsusedinthiswork.AsameasureofaccuracyforbothKrigingandlinearregression,weusetherootmeansquareerror(RMSE)whichgivesusanideaofhowgoodistheapproximationpredictionintheregionwherethesurrogatewastrained.TheRMSEisusuallyanexpensivewaytomeasureerrorbecauseweneedtestpointsthatwedonotusetoconstructthesurrogate.TheRMSEiscalculatedasfollows, RMSE=vuut mXi=1(^y(x(i)))]TJ /F4 11.955 Tf 11.95 0 Td[(yi)2 m;(5-1)where^yisthesurrogateapproximation,x(i)istheithtestpointofindependentvariables,yiistheresponseatthepointx(i),andmisthenumberoftestpointslocatedintheregionofinterest.Inthiswork,wecalculatedtheRMSEonlyintheregionswherethesurrogatemodelsweretrained.Polynomialswerechosenasbasisfunctionsforlinearregression.Thepolynomialdegreewasselectedtobetheonethatminimizesthecross-validationleave-one-out(CV-LOO)error.CV-LOOisoftenusedtomeasurethemodeladequacy.ThedegreethatwillminimizetheCV-LOOerrorwillvaryproblemtoproblemanditwilldependonthesampleanddomainsize,andthecharacteristicsofthefunctionbeingapproximated.Ingeneral,cross-validation(CV)erroriscalculatedbysplittingtheavailabledatapointsintoatrainingsetandatestset.Then,thesurrogatemodelisbuiltusingthetrainingsetandtheerrorismeasuredbycomparingthesurrogatevaluesagainstthetestset,typicallybyRMSE.Thetestingandtrainingpointsareexchangedsoattheendallthepointsbecometestingpointsonce.Inthiswork,weusedaparticularkindofCVerror,theCV-LOOerror,wherethetestsethasonlyonepoint,exceptforAPPapproachwhereweremovebesidesthepointitspermutations.ForfurtherdetailsaboutCV-LOO,thereadercanreferto[ 44 ].CV-LOOcanalsobeveryexpensivebecauseitbuildsthesurrogateasmany 127

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timesasavailabledatapoints.TheCV-LOOerrorcanbeexpressedasfollows, CV-LOO=vuut nuXi=1(^y)]TJ /F6 7.97 Tf 6.59 0 Td[(i(x(i)))]TJ /F4 11.955 Tf 11.95 0 Td[(yi)2 nu;(5-2)where^y)]TJ /F6 7.97 Tf 6.59 0 Td[(iistheapproximationtrainedwithouttheithtrainingpoint,yiistheresponseattheithtrainingpoint,andnuisthenumberoftrainingpoints.Thedeviationfromsymmetry(DS)ofthesurrogatemodelswasalsocomputed.TheDSofthesurrogatemodel^yindvariablesiscalculatedasthemeanofthesummationofthedierencebetweenthevalueatthepointianditsnspermutationpointssquared.Thiserrorequationcanbewrittenas, DS=vuuut 1 nonoXi=11 n(i)sn(i)sXj=1(^y(x(i)N0))]TJ /F1 11.955 Tf 12.75 0 Td[(^y(x(j)N(i)s))2;(5-3)wherenoistheoriginalnumberofdatapointsintheoriginalsetN0,n(i)sisthenumberofsymmetrypointsassociatedwiththeoriginalithpoint,x(i)N0istheithdatapointoftheoriginaldatasetN0,andx(j)N(i)sisthejthfreelyavailablesymmetrydatapointassociatedwiththepointi. 5.5SurrogateImplementation 5.5.1KrigingKrigingassumesthatthedatapointsaresampledfromanunknownfunctionthatobeyssimplecorrelationrules.NormallyKrigingisusedwiththeassumptionthatthereisnonoisesothatitinterpolatesexactlythefunctionvalues.Letusconsideraddimensionalfunctiony(x)suchthatx=(x1;x1;:::;nd).Krigingprediction,^y,isusuallymodeledasthesumofatrendfunction(x)andadiscrepancyfunctionZ(x) ^y(x)=(x)+Z(x):(5-4)Inparticular,weuseordinaryKriging,whichconsiderstobeconstantandZtobewrittenasasummationofcoecientstimesthebasisfunctions.Therefore,Eq.( 5-4 )can 128

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berewrittenas ^y(x)=+noXi=1b(i)(i)(x);(5-5)where(i)istheKrigingbasisfunctioncenteredattheithoriginaldatapoint,b(i)isthecoecientassociatedwiththeithoriginaldatapoint,andnoisthenumberoforiginaldatapoints.Gaussianbasisfunctionsoftheform (i)(x)=exp()]TJ /F6 7.97 Tf 17.74 14.94 Td[(dXj=1jx(i)j)]TJ /F4 11.955 Tf 11.96 0 Td[(xj2)(5-6)wereused,wherejisthehyperparameterofthejthvariableandx(i)jisthejthvariableoftheithoriginaldatapoint.Inthispaper,theKrigingapproximationswereperformedusingViana'sMATLABSurrogateToolbox[ 270 { 275 ].Thehyperparameterjwaslimitedtotherange[0.1,10]toavoidover-ttingorunder-tting.Inthispaper,threeoptionsareproposedtotakeadvantageofsymmetrieswithoutmodifyingtheinternalKrigingmachinery:APP,RD,andOS.CautionisneededwhenconsideringRDusingKrigingsurrogatesbecauseitmayleadtooverttingnoiseandtoill-conditioningofthecorrelationmatrixduetotheclosenessofpoints.However,thisdoesnotapplytoregressionsurrogates.SBF,whichrequiresmodifyingtheKrigingsourcecode,wasnotimplementedinthispaper,neverthelessweshowbelowhowKrigingsystemofequationscouldbemodiedincasethatthereaderwanttoimplementit.Considerafunctiony(x1;x2;x3:::;xd)suchthatthevariablepermutationsdonotmatter,i.e.y(x1;x2;x3;:::;xd)=y(x1;x3;x2;:::;xd)=y(x2;x1;x3;:::;xd)andsoon.Notethatweenforcetheconditionj=8j2f1;2;:::;dg.But,thisconstraintisonlynecessary,butnotsucient.Theconditionthattheamplitudesofallthebasisfunctionsassociatedwiththen(i)ssymmetrypointsarethesameneedstobealsoimposed.Sothebasisfunction(i)l(x)correspondentwiththelth 129

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symmetrypointassociatedwiththeithoriginaldatapointcanbeexpressedas (i)l(x)=exp()]TJ /F4 11.955 Tf 9.3 0 Td[(dXj=1x(i)j;l)]TJ /F4 11.955 Tf 11.96 0 Td[(xj2);(5-7)wherel=1;2;3;:::;n(i)s,andx(i)j;listhevalueofthejthvariableatlthsymmetrypointassociatedwiththeithoriginalpoint.Thefunctionytakesthesamevalueattheithpointandatitsn(i)spermutationpointstherefore,theysharethecoecientbifromEq.( 5-5 ).Eq.( 5-5 )canberewrittenas ^y(x)=+noXi=1b(i)0@(i)(x)+n(i)sXl=1(i)l(x)1A:(5-8) 5.5.2LinearRegressionLinearregressionisawidelyusedsurrogatemodelduetoitssimplicityandrobustness.UnlikeKriging,linearregressiondoesnotinterpolatethedatapointsanditisveryrobustfornoisyproblems.Inlinearregressionthefunctionshapeisassumedapriori,usuallybeingpolynomials.Asthenameindicates,linearregressionassumesalinearrelationshipbetweenthebasisfunctionandthecoecientsfoundbythet.Ingeneral,thelinearregressionmodelprediction^yofthefunctionycanbewrittenas ^y(x)=Xibii(x);(5-9)whereiarethelinearregressionbasisfunctions,xisthevectorofthefunctionvariables,andbiarethecoecientstobedetermined.Now,letusassumethatlinearregressionbasisfunctionsarethemonomialsofapth-degreepolynomial.Inthiswork,webuiltlinearregressionmodelsforuptothreevariablesusingpolynomialbasisfunctionsuptoa3rd-degreepolynomial.Thedegreeofthepolynomialselectedistheonethatminimizesthecross-validationleave-one-out(CV-LOO)error.Letusconsiderthenathree-variablefunctiony(x1;x2;x3)tobeapproximatedusinga3rd-degreepolynomial.Now,Eq.( 5-9 ) 130

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canbewrittenas^y(x)=b11+b2x1+b3x2+b4x3+b5x1x2+b6x1x3+b7x2x3+b8x21+b9x22+b10x23+b11x1x2x3+b12x21x2+b13x1x22+b14x21x3+b15x1x23+b16x22x3+b17x2x23+b18x31+b19x32+b20x33: (5-10)Weproposedanapproachtoincludeparametricsymmetriesinsurrogatemodelsmodifyingthelinearregressionbasisfunctions,wehavecalleditSBFanditisdescribedbelow.Considery(x1;x2;x3)tobethefunctionofinterestwherey(x1;x2;x3)=y(x1;x3;x2)=y(x2;x1;x3)andsoon.Therefore,wecanrewriteEq.( 5-10 )as^y(x)=~b11+~b2(x1+x2+x3)+~b3(x1x2+x1x3+x2x3)+~b4(x21+x22+x23)+~b5x1x2x3+~b6(x21x2+x1x22+x21x3+x1x23+x22x3+x2x23)+~b7(x31+x32+x33); (5-11)where~b1=b1,~b2=b2;b3;b4,~b3=b5;b6;b7,~b4=b8;b9;b10,~b5=b11,~b6=b12;b13;b14;b15;b16;b17,and~b7=b18;b19;b20.NotethatwhileEq.( 5-10 )has20coecientstobedetermined,Eq.( 5-11 )hasonlyseven.Thissameanalysiscanbeeasilyextendedtohigherorderpolynomialsandalargernumberofvariables. 5.6AnalyticalExample 5.6.1ProblemDescriptionToshowthereadertheoutcomeofimposingsymmetriesinsurrogatemodelsusingtheproposedoptions,weincludedananalyticalexample.Twodierentsurrogatemodelswerebuilt,Krigingandlinearregression,exploringthebenetsanddrawbacksofthefouroptionsdescribedinSection 5.4 .Wehavechosentoapproximatethefollowingtwovariateanalyticalfunction, y(x1;x2)=sinp x21+x22 p x21+x22+0:1cos2arctanx2 x1+ 2;(5-12)wherex1andx2aretheindependentvariables.Anadvantageoftakingabivariatefunctionisthatwecanplotit,alongwithitsapproximations,in2Dgureswhichareeasy 131

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tovisualize.Besidesitssmoothness,wechosethisfunctionbecauseitissymmetricaboutthex1=x2axis,i.e.y(x1;x2)=y(x2;x1),thereforewecantesttheproposedoptionsforimplementationofsymmetries. 5.6.2DesignofExperimentsTheareaofinterestistheuppermostrightquadrantDconsideringonly0x1;x212,i.e.D=fx1;x2j0x1;x212g.Weconsideragridofa49possiblepoints,i.e.77,wherex1andx2,inspiredbythecharacteristicsofourphysicalexample,canonlytakeintegervalues.TheRMSE(rootmeansquareerror,Eq.( 5-1 ))andtheDS(deviationfromsymmetry,Eq.( 5-3 ))werecalculatedinauniformgridof100100pointsinD.Wealsoassumedthatweknewapriorithatthemodeledfunctionpresentsreectionsymmetryaboutthex1=x2axis.InSection 5.6.3 wepresenttheresultsofapplyingtheoptionsAPP,RD,andOSusingKriging.InSection 5.6.4 wepresenttheresultsofapplyingtheoptionsAPP,RD,OS,andSBFusinglinearregression.Wecomparedtheperformanceoftheoptionstowardsthemeanperformanceof100dierentDoEsobtainedbyrandomlychoosingatrainingsubsetofno=28pointsoutofthetotalnt=49,asdescribedinSection 5.4 .Figure 5-3 shows3DandcontourplotsoftheanalyticalfunctiondescribedbyEq.( 5-12 )intheregionD.Filledcirclesrepresentsthesetofnttotalnumberofavailabledatapoints,Nt.Thecentraldashedlinerepresentsthesymmetryaxisx1=x2considered. 5.6.3KrigingResultsThreealternativeswereexploredAPP,RD,andOSwiththeaimofimprovingKrigingapproximationsusingtheparametricsymmetriesinherenttoourproblem.AswementionedinSection 5.5.1 ,wewillnotmodifyKrigingequationstherefore,SBFapproachwillnotbetested.Table 5-1 showsKriginghyperparameters1and2fortheapproximations^y,^yOS,^yRD,and^yAPPofthefunctiony(x1;x2).^yvaluesaregivenintriadswheretherst,secondandthirdnumbercorrespondtothemean,theminimumandmaximumvalues,respectivelyofthe100randomDoEs. 132

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A BFigure5-3. AnalyticalfunctiondescribedbyEq.( 5-12 ).A)Threedimensionalrepresentationoftheanalyticalfunction;B)Contoursoftheanalyticalfunction.Filledcirclesarethesetoftotalnumberofavailablepoints,Nt.Thecentraldashedlinerepresentsthesymmetryaxisx1=x2considered. Table 5-2 presentsRMSEandDSforKrigingapproximations^y,^yOS,^yRD,and^yAPPofthefunctiony(x1;x2)computedintheregionofinterestD.Theapproximation^ydoesnottakeintoaccountanysymmetryofthefunctionandusesonly28trainingpoints.^yresultsaregivenintriadswheretherst,secondandthirdvaluesarethemean,minimumandmaximumvaluesof100randomDoEsof28pointsoutofthe49availableintotal.Krigingapproximationsof^yOS,^yRD,and^yAPParealsoshown.ThepercentagescomparetheRMSEandDSofeachapproximationwithrespecttothemeanapproximation.NotealsothatthereisalargespreadoftheaccuracywithdierentDoEswhenthesymmetryisnotusedFigure 5-4 complementsTable 5-2 informationshowingtheRMSEforeachofthe100randomDoEsalongwiththeRMSEvaluesforAPP,RDandOSapproaches.SummarizingtheinformationinTable 5-2 ,weconcludethatimposingsymmetriesleadstoanRMSEreductionnomatterwhichofthethreeapproachesisused.Inthiscase,thebestperformanceisachievedby^yAPPwitha97%RMSEreductioninD. 133

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Table5-1. Kriginghyperparameters1and2for^y,^yOS,^yRD,and^yAPPoftheanalyticalfunction.Thetriadsfor^yarethemean,minimum,andmaximumvalues,respectivelyofthe100DoEs Model12 ^yOS0.350.23^yRD0.310.63^yAPP1.251.25 Table5-2. RMSEandDSforKrigingapproximations^y,^yOS,^yRD,and^yAPPoftheanalyticalfunctioncomputedintheregionofinterestD.Thetriadsfor^yarethemean,minimum,andmaximumvalues,respectivelyofthe100DoEs.ThepercentagescomparetheRMSEandDSofeachapproximationwithrespecttothemeanapproximation ModelRMSEDS ^y[0.117,0.086,0.160](0.057,0.021,0.133)^yOS0.086(74%)0.05(88%)^yRD0.024(21%)0(0%)^yAPP0.004(3%)0(0%) Figure5-4. Krigingsurrogateerrors:crossesrepresenttheRMSEforeachoneofthe100DoEsof28pointsoutofthe49.Themeanofthe100DoEsRMSEishighlighted.TheRMSEforeachapproachisalsonoted 134

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ComputingthefreeRMSEallowedustoobtain^yOS.ThispredictioncorrespondstotheminimumfreeRMSEcomputedusingpermutationpointswhichwillbeavailabletotheuser.Figure 5-4 showsagoodagreementbetweenthefreeRMSEandtheRMSEcomputedusingtestpoints,asyoucansee^yOSoverlapsthecrosswiththelowestRMSEinFigure 5-4 .Figure 5-5 showstheresultingcontourplotsoftheKrigingapproximationsAPP,RDandOSoftheanalyticalfunctionwhoseerrorsareshowninTable 5-1 andinFigure 5-4 .FilledcirclesrepresentNu,thesetofnutrainingpointsusedineachcase.TheOSapproachdoesnotmaintainsymmetriesaswecanseeinFigure 5-5A .ThemaindierencebetweenRDandAPPapproachesliesinthelineofsymmetry.Byconstruction^yRDapproximationpresentsacuspatx1=x2line,whichisevidentinFigure 5-5B .Ontheotherhand,Figure 5-5C showsthatfor^yAPPwegetasmoothfunctionwithzeroslopenormaltothex1=x2line,asintheanalyticalfunction.^yRDand^yAPPgive,byconstruction,asymmetricsurrogate,i.e.^y(x1;x2)=^y(x2;x1)8fx1;x2g.Ontheotherhand,for^yOSthisisnotthecase,givingadeviationfromsymmetry,DS,of0.05.Figure 5-6 showstheabsoluteerrorbetweentheanalyticalfunctionandtheapproximationsinD.Overall,Krigingdoesagoodjobttingthisfunctionwithamaximumrangenormalizedabsoluteerrorof8%forOS,whichistheonewiththepoorestperformance.Clearly,thebestperformanceisachievedbyAPP. 5.6.4LinearRegressionResultsInthissection,weexplorefouroptionsofimposingsymmetriesusinglinearregression:APP,SBF,RD,andOS.OntopoftheapproximationsexploredintheprevioussectionusingKriging,herewealsoproposetomodifythelinearregressionbasisfunctionstoimposethesymmetries,whichwehavecalledtheSBFapproach(Section 5.5.2 ).Thelinearregressionsurrogatewasbuiltusinga3rd-degreepolynomialbecauseitminimizestheCV-LOOerrorforthiscase. 135

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A B CFigure5-5. ContourplotsoftheKrigingapproximationsAPP,RDandOS.Filledcirclesrepresentthenutrainingpointsusedineachcase.A)^yOScontours;B)^yRDcontours;C)^yAPPcontours. Table 5-3 showstheRMSEandtheDSinthedomainDfortheproposedapproximations.Again,theapproximations^ydonottakeintoaccountanysymmetryofthefunctionandusesonlythe28trainingpoints.^yresults,showninTable 5-3 ,arepresentedintriadswheretherst,secondandthirdvaluesarethemean,minimumandmaximumvaluesof100randomDoEsof28pointsoutofthe49available.Theerrorsoftheotherapproximationsinthetablearecomparedtothemeanusingpercentvalues. 136

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A B CFigure5-6. ContourplotsoftheabsoluteerrorcontoursoftheKrigingapproximationsAPP,RDandOS.ThemaximumabsoluteerrorisobtainedusingOSandrepresentsan8%ofthefunctionrangeinD.A)^yOSabsoluteerrorcontours;B)^yRDabsoluteerrorcontours;C)^yAPP=SBFabsoluteerrorcontours. InFigure 5-7 theRMSEofeachofthe100DoEsisrepresentedbyacross,theirmeanisalsohighlighted.Fortheotherapproximations^yOS,^yRD,and^yAPP=SBFwepresenttheirsingleRMSEvalue.Table 5-3 showsthatoveralllinearregressionislesspronetoimprovementsduetoimposedsymmetriesthanKriging.Itmaybethatbecausethefunctionhaslargegradientsneartheorigin,polynomiallinearregression,propagatestheseglobally.Ontheotherhand,forKrigingtheglobalbehaviorisnotasaected;soitresultsinalocallyimprovedsurrogateneartheaddedpointsandadecentperformancefarfromthem. 137

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Table5-3. RMSEandDSerrorsforlinearregressionapproximations^y,^yOS,^yRD,^ySBF,and^yAPPoftheanalyticalfunctioncomputedintheregionofinterestD.Thetriadsfor^yarethemean,minimum,andmaximumvalues,respectivelyofthe100DoEswith28pointsofthe49ofthegrid.ThepercentagescomparetheRMSEandtheDSofeachapproximationwithrespecttothemeanapproximation ModelRMSEDS ^y(0.146,0.119,0.298)(0.101,0.023,0.779)^yOS0.124(85%)0.023(23%)^yRD0.113(77%)0(0%)^ySBF0.122(84%)0(0%)^yAPP0.122(84%)0(0%) Figure5-7. Linearregressionsurrogateerrors:crossesrepresenttheRMSEforeachoneofthe100DoEsof28pointsoutofthe49.Themeanofthe100DoEsRMSEishighlighted.TheRMSEforeachapproachisalsonoted 138

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A B CFigure5-8. ContourplotsofthelinearregressionapproximationsOS,RD,APPandSBF.FilledcirclesrepresentthenutrainingpointsusedineachcaseA)^yOScontours;B)^yRDcontours;C)^yAPPcontours. Ifthefunctionhasimportantvariations,asisthecaseofthisanalyticalfunctionneartheorigin(Fig. 5-3 ),linearregression,whichisbasedonpolynomials,translateslocaluctuationsinglobaluctuations.Ontheotherhand,forKrigingtheglobalbehaviorisnotasaected;soitresultsinalocallyimprovedsurrogateneartheaddedpointsandadecentperformancefarfromthem.Inotherwords,inlinearregressionwehavefewercoecientsthaninKrigingtobedeterminedhenceaddingmorepointsislessuseful.Figure 5-8 showsthecontourplotsofthelinearregressionapproximationsAPP,RD,OS,andSBFoftheanalyticalfunctionwhoseerrorsareshowninTable 5-3 andin 139

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A B CFigure5-9. ContourplotsofthelinearregressionapproximationsOS,RD,APPandSBF.Filledcirclesrepresentthenutrainingpointsusedineachcase.A)^yOSabsoluteerrorcontours;B)^yRDabsoluteerrorcontours;C)^yAPP=SBFabsoluteerrorcontours. Figure 5-7 .Figure 5-9 showstheresultingcontourplotsoftheabsoluteerrorcontoursforAPP,RD,OS,andSBFlinearapproximationsoftheanalyticalfunction.ThemaximumabsoluteerrorisobtainedusingOSandrepresentsa30%ofthefunctionrangeinD.Althoughthe^yRDcuspatx1=x2lineisevenmoreevidentthanforKriging(Figure 5-8B ),thisistheapproachthatachievesthelowestRMSEinD(77%)andbetterperformancethanallthe100DoEs.Thissuccessof^yRDisexpectedbecausetheweightontheborderisreducedintheotheroptions.^ySBFand^yAPPapproximationsareidenticalbecauseforlinearregressionaddingallthepermutationpointsastrainingpoints 140

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andimposingsymmetricbasisfunctions,areequivalent.Aswementionedbefore,^yOSsuccessfullypredictedtheDoEwitharelativelylowRMSE(13thofthe100),asshowninFigure 5-7 .ThisisstillasuccessfulpredictionreinforcingthefactthatthepermutationpointsdoagoodjobastestpointsinthefreeRMSEcalculation.Finallynotethat^yRD,^ySBFand^yAPPgive,byconstruction,asymmetricsurrogate. 5.7PhysicalExample 5.7.1ProblemDescriptionThecomputationaldomainiscomposedofa0.76cmdiameterinnercirclecontaininghot,high-pressuregaswhichissurroundedbya10cmdiameterannulusofairandglassparticlemixture(5%particles,95%air).The120cmdiameteroutermostannuluscontainsambientair.Exceptfortheinnercircle,therestofthedomainisinitiallyunderstandardconditionsofpressureandtemperature.Figure 4-2 isaschematicofthecomputationaldomain(nottoscale).Thepolargridusedhas36,096computationalcellsand31,250computationalparticles.Formoreinformationaboutthephysicalexampledescription,computationalapproach,andmodelingreferto[ 5 ].Between[ 5 ]andthepresentworktherearetwodierences1)inourpapertheuxschemewasupdatedfromAUSM+[ 276 ]toAUSM+-up[ 245 ],and2)wepresentresultswheretheseedfortheparticlesinitiallocationismaintainedconstanthencesameparticleinitialconditionsleadstosameoutputs.Thebaseparticlevolumefraction(PVF)wassetto5%,whichisrelativelylow,toavoidtheeectsofcompactingparticles.Theouterannuluscontainstheblastwaveduringtheentiresimulationtime,500s.TheperturbationsimposedtothePVFareinspiredby[ 5 ].ThebasePVFwasperturbedusingasuperpositionofuptothreesinusoidalwaves.Eq.( 5-13 )showsthemathematicalexpressionoftheperturbationwhileEq.( 5-14 )showstheassociatedenergyconstraintofp 2=100(14%)basedon[ 6 ].NotethatthePVFisinitiallyconstantintheradialdirectionandtheperturbationisrestrictedtothecircumferentialdirection. p()=p0[1+A1cos(k1+1)+A2cos(k2+2)+A3cos(k3+3)];(5-13) 141

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Figure5-10. Schematicofthecomputationaldomain(nottoscale) subjectto q A21+A22+A23=p 2=100;(5-14)wherepisthePVFatagiven,andp0istheconstantbasePVF.Modeamplitudes(A1,A2andA3),phases(1,2and3)andwavenumbers(k1,k2,andk3)aretheperturbationparameters.Furthermore,withoutlossofgeneralitywecantake1=0andonlythephaseoftheothertwomodeswithrespecttotherstonematters.Toillustratethenatureofinitialperturbation,Figure 4-6 comparesPVFcontoursfortheunperturbedcasewithacasewhereatri-modalperturbationisimposed.Wedividethedomainintocircularsectorswithidenticalvolume.Inourproblem,thevolumeofthesectorsremainsconstantandequaltohR2=N.Here,R=0.6mistheradiusofthecomputationaldomainshowninFigure 4-2 ,h=0.02misthecellheight,andN=256isthenumberofgridcellsintheazimuthalcoordinate.Inthispaper,wepresenttwometrics.Therstoneiscalledandisasimplicationofthesecondone,.Themetricwasintroducedtofacilitatethereader'scomprehensionanditconsistsofthenumberofparticlesinthesectorwithmostparticlesasafunction 142

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A BFigure5-11. contoursatinitialtime.A)PVFcontoursatinitialtimefortheunperturbedcaseA1=A2=A3=0;B)PVFcontoursatinitialtimefortheperturbedcasewhereA1=A2=A3=p 2=300,k1=10,k2=14,k3=17,and1=2=3=0. oftime.Thenumberofparticlesinthesectorwiththemostparticles,,asafunctionoftimeiscalculatedas (t)=max 1 VolpXrPVF(;r;t)Volc(r)!;(5-15)wheretistime,rtheradialcoordinate,andistheangularcoordinate,Volpisthevolumeofasingleparticle,whichisconstantthroughoutthesimulation,Volcisthevolumeofacellwhichchangesasafunctionofrduetothepolargridused,andPVF(;r;t)isthelocalparticlevolumefractionatthegridcell.Thesecondmetricismorecomplexanditwaschosenaftersubstantialanalysisandnumericalexperimentationasagoodmetricofthedeparturefromaxisymmetry.Thenormalizedparticlevolumedierence,,asafunctionoftimeiscalculatedas (t)=f(t) f(t=0);(5-16) 143

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wherefiscalledparticlevolumedierenceanditiscomputedas f(t)=max(f(;t)))]TJ /F1 11.955 Tf 11.96 0 Td[(min(f(;t));(5-17)wheremax(f(;t))istheparticlevolumefofthesectorwithmostparticles,andmin(f(;t))istheparticlevolumefofthesectorwiththeleast.Theparticlevolumef(;t)isthevolumeofparticlesinasectorwhichdependsonthetimetandtheangularcoordinate.Notethat(t)=max(f(;t))=Volp. 5.7.2DesignofExperimentsInwhatfollowswewillrestrictattentiontoasubsetofatri-modalperturbationwheretheamplitudesofthethreemodesarechosentobeequalwhich,alongwiththeenergyconstraint,yieldsA1=A2=A3=A=p 2=300,andzerophasedierencebetweenthethreemodes,i.e.,1=2=3=0.WiththesesimplicationsEq.( 5-13 )becomes p()=p0f1+A[cos(k1)+cos(k2)+cos(k3)]g:(5-18)AscanbeseeninEq.( 5-18 )theorderofk1,k2,andk3doesnotmatter.Thecasewithwavenumbers(k1,k2,k3)anditsvepermutations(k1,k3,k2),(k2,k1,k3),(k2,k3,k1),(k3,k1,k2),(k3,k2,k1)willgiveexactlythesameoutput.ThisisknownasthedihedralorD3symmetrygroupandwewillrefertothesixpointsthatareconnectedbythedihedralsymmetryasthepermutationpoints.Thus,inthek1;k2;k3space,resultsfromasimulationwithoneparticularvalueofthesethreewavenumbers,automaticallyyieldsresultsfortheothervecombinationsofwavenumbers.Inconstructingasurrogatethatspansthek1,k2,andk3,wenotonlyusetheresultsoftheoriginalsetofsimulationsbutalsothepermutationpoints.AsintheanalyticalexamplepresentedinSection 5.6 ,thetrainingdatasetiscontainedinaregionofinterestDandinthiscaseD=fk1;k2;k3j1k1;k2;k325g.Figure 5-12 andFigure 5-13 representthelocationandvalueofdierentdatasetscorrespondingtothemetricand,respectively.Whilethelocationofthepointsis 144

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identicalinbothgures,thecolorrepresentingthevalueofeachmetricvaries.Thegeneralbehaviorofthemetricsisquitesimilar,andoverallwecansaythattheirvalueincreasesasthewavenumbersvalueincrease.TheoriginaltrainingdatasetNo(no=34points)isplottedinFigure 5-12A forandinFigure 5-13A for,andcanbefoundinSection 5.9 .Figure 5-12B showsthe34datapointsinFigure 5-12A ,withdatapointsoutsidetheregionk1k2k3replacedbytheirpermutationpointsinsidetheregion.Figure 5-13B showsthe34datapointsinFigure 5-13A restrictedtotheregionk1k2k3.Figure 5-12C showsthepermutationpointstogetherwiththeoriginaldatapoints(nt=204=343!)for.Figure 5-13C showsthepermutationpointstogetherwiththeoriginaldatapoints(nt=204=343!)for.Here,asintheanalyticalexample,wetrainedtwosurrogates,Krigingandlinearregression.Themaindierencebetweenthephysicalexampleandtheanalyticalexampleisthepresenceofnoiseinthephysicalexample.Whilethedataintheanalyticalexamplehasnonoise,thedatainthephysicalexamplehasnumericalnoiseresultingfromthenonlinearsimulation.Althoughidenticalsimulationswillgivethesameresults,theparticlesareinitiallyrandomlyplacedwithinthecomputationaldomainanddierentrandominitialdistributionsleadtoslightlydierentresults.Noiseinvestigationrevealedthatthemagnitudeofthenoiseisabout1%ofthettedfunctionvalues[ 277 ].ThereareversionsofKriging,suchasKrigingwithnugget[ 278 ][ 279 ][ 280 ],thattakeintoaccountnoise.However,withdataobtainedfromsimulations,itiscommontouseinterpolatingKrigingbecauserepeatedsimulationsproduceidenticalresults.InSection 5.7.3 wepresenttheresultsofapplyingOS,RD,andAPPoptionsinbuildingKrigingsurrogate.InSection 5.7.4 wepresenttheresultsofapplyingOS,RD,SBF,andAPPoptionsinbuildingalinearregressionsurrogate.Thelinearregressionapproximationswereperformedusinga2rd-degreepolynomialwhichminimizestheCV-LOOerrorforthephysicalexample. 145

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A B CFigure5-12. Setofthenutrainingdatapoints,Nu,usedtotrainthesurrogatesforeachcasetoapproximate.A)Oneofthe100DoEsof34datapointsusedtoconstruct^;B)Permutationpoints,fromtheoriginalset,restrictedtotheregionk1k2k3usedtobuild^RDand^SBF(34datapoints);C)PermutationpointsinDtogetherwiththeoriginaldatapoints(204=343!datapoints)usedtobuild^APP. WeusetheRMSEastheglobalmeasureoferrorinD.TheRMSEwasperformedusing343!=204testdatapointsthatwerenotusedtotrainthesurrogate.ThetestdatapointswereincludedinSection 5.9 5.7.3KrigingResultsTable 5-4 showsKriginghyperparameters1,2and3fortheapproximations^,^OS,^RDand^APPusingKrigingsurrogate.^valuesaregivenintriadswheretherst,second 146

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A B CFigure5-13. Setofthenutrainingdatapoints,Nu,usedtotrainthesurrogatesforeachcasetoapproximate.A)Oneofthe100DoEsof34datapointsusedtoconstruct^;B)Permutationpoints,fromtheoriginalset,restrictedtotheregionk1k2k3usedtobuild^RDand^SBF(34datapoints);C)PermutationpointsinDtogetherwiththeoriginaldatapoints(204=343!points)usedtobuild^APP. andthirdvaluescorrespondtothemean,theminimumandmaximum,respectivelyofthe100randomDoEsofno=34datapointsoutofnt=204.Table 5-5 showstheRMSEandDSforKrigingapproximationsof.^valuesaregivenintriadswheretherst,secondandthirdvaluesaretheaverage,theminimum,andthemaximumvaluesfoundreceptively,amongthe100randomDoEsofno=34pointsoutofnt=204.^OSRMSEiscalculatedusingtheoptimalsetfromtheprevious100 147

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DoEs,i.e.theDoEwithlowestfreeRMSE.^RDistrainedintheregionk1k2k3andthentheapproximationisreectedwithrespecttotheplaneki=kjwithi;j=1;2;3andi6=j.Finally,^APPRMSEistheresultofaddingallthentpermutationpointsastrainingpoints. Table5-4. Kriginghyperparameters1,2and3for^,^OS,^RDand^APPforthephysicalexample.Thetriadsfor^arethemean,minimum,andmaximumvalues,respectivelyofthe100DoEsof34datapointsoutof204 Model123 ^(1.1,0.1,8.0)(1.8,0.1,10.0)(2.0,0.1,10.0)^OS0.40.40.2^RD10.03.53.1^APP5.34.65.4 Table5-5. RMSEandDScalculatedinthedomainofinterestDfor^,^OS,^RDand^APPusingKrigingsurrogate.Thetriadsfor^arethemean,minimum,andmaximumvalues,respectivelyofthe100DoEsof34datapointsoutof204.ThepercentagescomparetheRMSEandDSofeachapproximationwithrespecttothemean ModelRMSEDS ^(5556,3962,8776)(4326,2301,7160)^OS3962(71%)2974(69%)^RD5827(105%)0(0%)^APP5313(96%)277(6%) Table 5-5 alongwithFigure 5-14 showthatOSapproachsuccessfullyfoundtheDoEwithlowestRMSEbycalculatingthefreeRMSE.With^OStheRMSEimproved29%withrespecttothemean,whichistheoptionthatperformedthebestinthiscase.Whiletheimprovementwithrespecttothemeanofthe100DoEsisonly29%,theimprovementwithrespecttotheworstcaseismorethan50%.AsTable 5-4 shows,for^RDand^APP,theKrigingoptimizerfoundhigheris.Thehyperparameteriisdirectlyrelatedtohowcorrelatedthedataisintheapproximation.Asigetshigher,thelessapointwillinuenceanotherpointinitsneighborhood.Ahighiissometimesduetopooroptimizationoritcanalsoreectttingthenoise.For^APPhigheriistheoutcomeofapooroptimizationresultingfromhavingtoomanypoints. 148

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Figure5-14. Krigingsurrogateerrors:crossesrepresenttheRMSEforeachoneofthe100DoEof34datapointsoutof204.Themeanofthe100DoEsRMSEishighlighted.TheRMSEcorrespondingtoeachapproachisalsonoted Ontheotherhand,for^RDthedatapointswerepackedmorecloselyhencethenoiseinttingdataisaccentuatedandtherefore,wegethigheri.Ontopofhighi,wealsoobservethat1isthreetimesbiggerthan2and3.InSection 5.5.1 weshownthat1,2,and3shouldtakethesamevalue,hence^RDisthepoorestapproximation.Thispoorestapproximationisstillbetterthantheworst32casesoutofthe100DoEs.^RDgivesbyconstructionasymmetricsurrogate(DS=0),ontheotherhand,theDSpredictionfor^OSis2974whilefor^APPis277.Surprisingly,although^OShasthelargestdeparturefromsymmetry(i.e.thehighestDS),contrarytotheanalyticalfunction,isthebestapproximationamongthethreeonesconsidered.Table 5-6 andTable 5-7 showthesameinformationshowninTable 5-4 andTable 5-5 ,respectively,butforthemorecomplexmetric,.Inthiscase,thebestperformanceisachievedby^APPreducingtheRMSEby27%.OSapproximationfoundthe17thlowestRMSEamongthe100.ThepoorresultsforRDreectthefactthat1,2,and3,based 149

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Table5-6. Kriginghyperparameters1,2and3for^,^OS,^RDand^APPforthephysicalexample.Thetriadsfor^arethemean,minimum,andmaximumvalues,respectivelyofthe100DoEsof34datapointsoutof204 Model123 ^(0.8,0.2,3.7)(1.1,0.2,10.0)(1.4,0.1,10.0)^OS0.30.70.4^RD6.60.11.0^APP2.52.83.5 Table5-7. RMSEandDScalculatedinthedomainofinterestDfor^,^OS,^RDand^APPusingKrigingsurrogateforthephysicalexample.Thetriadsfor^arethemean,minimum,andmaximumvalues,respectivelyofthe100DoEsof34datapointsoutof204.ThepercentagescomparetheRMSEandDSofeachapproximationwithrespecttothemean ModelRMSEDS ^(0.132,0.093,0.209)(0.120,0.069,0.213)^OS0.113(86%)0.081(68%)^RD0.131(99%)0(0%)^APP0.096(73%)0.016(13%) Figure5-15. Krigingsurrogateerrorsfor^:crossesrepresenttheRMSEcorrespondingtoeachoneofthe100DoEsof34datapointsoutof204.Themeanofthe100DoEsRMSEishighlighted.TheRMSEcorrespondingtoeachapproachisalsonoted 150

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ontherestricteddomain,aresodierent.However,^RDstillperformsbetterthan47casesoutofthe100.^RDgivesbyconstructionasymmetricsurrogate(DS=0).Ontheotherhand,for^OSand^APP,theDSpredictionsare0.081and0.016,respectively. 5.7.4LinearRegressionResultsTable 5-8 andTable 5-9 showtheRMSEforthelinearregressionapproximationsofand,respectively.^and^valuesarepresentedintriadswheretherst,secondandthirdvaluesarethemean,minimumandmaximum,respectivelyof100randomDoEsofno=34datapointsoutofnt=204.ThetablesalsoshowtheRMSEandDSoftheoptions:OS,RD,SBF,andAPP.ThepercentagesinTable 5-8 andinTable 5-9 compareeachapproximationwiththemean. Table5-8. RMSEandDScalculatedinthedomainofinterestDfor^,^OS,^RD,^SBFand^APPusingalinearregressionsurrogateforthephysicalexample.Thetriadsfor^arethemean,minimum,andmaximumvalues,respectivelyofthe100DoEsof34datapointsoutof204.ThepercentagescomparetheRMSEandDSofeachapproximationwithrespecttothemean ModelRMSEDS ^(3926,3541,5427)(1655,787,3452)^OS3742(95%)1754(106%)^RD3364(86%)0^SBF3636(93%)0^APP3636(93%)0 Table 5-8 showsthatthehighestimprovementisachievedusingthe^RD,whichreachesanRMSEreductionof14%withrespecttothemean.Theapproximations^SBF,^APPand^OSshowsmallRMSEreductionwithrespecttothemean,however,theRMSEreductionwithrespecttotheworstcaseisatleast31%.Thepoorestperformancewasachievedby^OShowever,itfoundtheDoEwiththe26thlowestRMSEamong100.^RD,^SBFand^APPgive,byconstruction,asymmetricsurrogate(DS=0),ontheotherhand,for^OStheDSpredictionis1754.Table 5-9 showsthatthebestperformanceisachievedby^SBFand^APPreachinga9%RMSEreduction.^OSfoundthe25thbestRMSEamongthe100.^RDshows 151

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Figure5-16. Linearregressionsurrogateerrorsfor^:crossesrepresenttheRMSEforeachoneofthe100DoEsof34datapointsoutof204.Themeanofthe100DoEsRMSEishighlighted.TheRMSEforeacheachapproachisalsonoted. Table5-9. RMSEandDScalculatedinthedomainofinterestDfortheapproximations^,^OS,^RD,^SBFand^APPusingalinearregressionsurrogate.^valuesarepresentedintriadsbeingtherst,thesecond,andthethirdnumbersthemean,theminimumandthemaximum,respectivelyof100DoEsof34pointsoutof204.ThepercentagescomparetheRMSEandDSofeachapproximationwithrespecttothemean ModelRMSEDS ^((0.086,0.009,0.113)(0.043,0.006,0.107)^OS0.080(93%)0.006(14%)^RD0.091(106%)0^SBF0.078(91%)0^APP0.078(91%)0 152

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Figure5-17. Linearregressionsurrogateerrorsfor^:crossesrepresenttheRMSEforeachoneofthe100DoEsof34datapointsoutof204.Themeanofthe100DoEsRMSE,^,ishighlighted.TheRMSEforeachapproachisalsonoted. anRMSEincreaseof6%withrespecttothemeanhowever,itperformedbetterthantheworst25casesoutof100.^RD,^SBFand^APPgive,byconstruction,asymmetricsurrogate(DS=0)whilefor^OStheDSdiscrepancyis0.006. 5.8ConcludingRemarksWhenweconstructsurrogatemodelsforaproblemthatpresentsparametricsymmetriesweobtainfreeadditionaldatapoints.Astraightforwardapproachistoaddthesefreelyavailabledatapointstotheoriginalsettogetthedesignofexperimentsforbuildingthesurrogatemodel.Surprisingly,thereareotheroptionsthatareoftenmoreaccurate.Inthiswork,wepresentedotherthreeoptionstoincludethesymmetriesinherenttoaproblemwhilebuildingsurrogatemodels.Twoofthemarestraightforwardwhileoneisoriginalandrequiresamoredetaileddescription.Twodierentsurrogatemodelswereconsidered:Krigingandlinearregression.Resultsforananalyticexample,andfortwo 153

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functionsderivedfromaphysicalproblemwereshown.Themaindierencebetweentheanalyticalexampleandthephysicalexampleisthatthephysicalexamplehas1%noiseduetonumericsinnonlinearsimulations.Wecomparedtheperformanceofthepresentedoptionstoincludethesymmetriesagainst100randomdesignofexperimentswherenosymmetrieswereimposed.Wefoundthateachoftheoptionstoincludethesymmetriesperformedthebestinoneormoreofthestudiedcases.Wealsofoundthatimposingsymmetrieskeptusfarfromthedesignofexperimentswiththepoorestperformance.Fortheanalyticalexample,allthepresentedoptionsperformedbetterthanatleastthe97caseswiththepoorestperformanceoutof100.Forthephysicalexample,allthepresentedoptionsperformedbetterthanatleastthe25caseswiththepoorestperformanceoutof100. 154

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5.9Data Table5-10. Originalsimulationtrainingdatapointsforsurrogateconstructioninthephysicalexample k1k2k3(k1;k2;k3)(k1;k2;k3) 181202.55813237231101212.365172306506242.09229214368725152.877852486201216253.0051252838251582.90509249041241012.3589922854151172.15152229731424163.01002253175218222.55974237315813122.94052249801192562.7938824322016242.14688222383142262.7219824305265252.459272262631720103.151772555372015243.08202255453151252.7706524524592052.7421724296781892.861532468482521153.04366254188520182.857942465111411203.08009255875221542.85372251319121162.427923335012672.73006241280922112.9500624996951242.47839234362152252.53333234784165212.7639324187196152.85677244570258112.785982454141525203.0244725418821982.82646246679 155

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Table5-11. Originalsetofsimulationtestdatapointsusedtotesttheperformanceofthesurrogatemodelsinthephysicalexample k1Vk2Vk3V(k1V;k2V;k3V)(k1V;k2V;k3V) 2518193.0856425562243102.1766522373261712.19887224576516202.829952530912116133.0349625410349232.6796239846912162.95698251572242182.858282477761710143.05242254019111562.81022245836113142.71975231409245212.6837241364256132.80843246089192122.556223638772452.50458235037201482.937072590811920233.127762573095642.1867922432383212.430232331811318143.0973525655024562.43152224829624112.73843242967215102.3889123250615812.279732270231810223.03838254272213182.76936237990114172.57546234025621222.7468823621816182.2724522626335162.26977226432912222.953222514881017233.00322261190110142.367982313251218233.09109255875 156

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CHAPTER6METRICSTUDYInthischapter,IdescribeanddiscussindetailthetwometricsthatIusedtoquantifythedeparturefromaxisymmetryinthiswork.TherstmetricisdenotedbyFandIrefertoisastheNormalizedFourierEectivePerturbationmetric.Fisametricbasedonenergy,ittakesintoaccounteachofthemodalamplitudespresentintheparticlesdistribution(L2norm).ThesecondmetricisdenotedbyandIrefertoitastheNormalizedMaximumParticleVolumeDierence.onlytakesintoaccountthemaximumdierenceinparticlevolumebetweenradialsectorsinthecomputationaldomain(L1norm).Iaminterestedinmeasuringtheamplicationofthedeparturefromaxisymmetry.Inthiswork,Iammainlyinterestedinthisamplicationatthenaltime500s. 6.1TheNormalizedFourierEectivePerturbationInthissection,IdescribehowFisobtained.First,Idividethecomputationaldomainintoradialsectorswithidenticalvolume.Inourproblem,thevolumeofthesectorsremainsconstantandequaltohR2=N,whereR=0.6mistheouterradiusofthecomputationaldomain,h=0.02misthecellheight,andN=256isthenumberofgridcellsintheazimuthalcoordinate.Atanyangularcoordinateandsimulationtimet,wecancalculatetheparticlevolume,PV(;t).Becauseofthecylindricalnatureofthephysicalproblem,wehaveaninherentperiodicityin02.Therefore,wecanrewritePV(;t)as PV(;t)=N=2Xk=)]TJ /F6 7.97 Tf 6.59 0 Td[(N=2Akexpik2 N;(6-1)whereAkistheFouriercoecientcorrespondenttothekthmode.TheFouriercoecientsarecomplexandtheyaregivenbytheFouriertransform Ak(t)=1 NN)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xj=0PV(j;t)exp)]TJ /F4 11.955 Tf 9.3 0 Td[(ik2 Nj:(6-2) 157

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AplotofjAkjasafunctionofthewavenumberkgivesuswhatiscommonlycalledtheamplitudespectrumoftheperiodicsignalPV(;t).Figure 6-1A showsPV(;t=0)normalizedspectrumsquaredwhileFigure 6-1B showsPV(;t=500s)normalizedspectrumsquared.ThespectraarenormalizedbyjA0j2.BecausePV(;t)isarealperiodicfunctionitsspectrumiseven,thereforeIonlyshowitforthepositivewavenumbers. A BFigure6-1. PVamplitudespectrumsquaredforA1=0:13,A2=0:039,A3=0:039,k1=8,k2=17,k3=15,and1=0:200,2=0:520,3=4:851.ThespectrumisnormalizedbyjA0j2. IdenetheFouriereectiveperturbation~Fas ~F(t)=N)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xk=0A2k A20)]TJ /F1 11.955 Tf 11.95 0 Td[(1=N)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xk=1A2k A20:(6-3)ThenthenormalizedFouriereectiveperturbationFiscalculatedas F(t)=~F(t) ~F(t=0)=~F(t) 0:02:(6-4)Notethat~F(t=0)=0:02forperturbedcasesbecauseoftheenergyconstraintimposedontheamplitudes(Eq.( 4-21 )).AlsonotethatFatinitialtimeis1byconstructionforallcases.ForthecasepresentedinFig. 6-1 at500s(Fig. 6-1B )thevalueofFis6.25. 158

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6.2TheUnlteredNormalizedMaximumParticleVolumeDierenceInthissectionIdescribehowthemetricisobtained.Tocompute,thePV(;t)iscalculatedasexplainedinEq.( 6-1 )andiscalculatedasafunctionoftimeas (t)=PV(t) PV(t=0);(6-5)wherePViscomputedas PV(t)=max(PV(;t)))]TJ /F1 11.955 Tf 11.95 0 Td[(min(PV(;t));(6-6)wheremax(PV(;t))isthePVofthesectorwithmostparticlesattimet,andmin(PV(;t))isthePVofthesectorwithleastparticlesattimet.atinitialtimeisonebyconstructionforallcases.ForthecasepresentedinFig. 6-1 at500s(Fig. 6-1B )thevalueofis3.06. 6.3MetricsComparisonFirstnotethat,inthiswork,whenreferringtoametricwithoutspecifyingthetimeitcorrespondstothenaltime,500s.ThatisF(t=500s)=Fand(t=500s)=.IshowedinSection 4.7.4 thatthereisastrongdependencebetweenthekeffandF.Furthermore,asensitivityanalysismadeforshowedthatthephases1,2,and3havenegligiblesensitivityindex.InSection 4.7.3 itwasalsoshownthatthedependenceofFonthephasesisontheorderofthenoise(5%).Therefore,Ihavechosentostudyindetailsixcasesthathavesimilarkeff(Eq.( 4-22 )),whichcanbedescribedasanamplitudeaveragedwavenumberthatdoesnotdependonthephases.Table 6-1 showstheparametervaluesoftheperturbationsimposedineachcase.Thesinglemodalcase(SM)istheonlypossiblecasewithkeff=10ifweconsider1=0.Thechosenbimodalcase(BM)hasidenticalamplitudesandthephasesaresettozero.Fourtrimodalcaseswereselected,therstone(TM1)isthecasewhereeachmodehasidenticalamplitudesandthephasesaresettozero.Thesecondtrimodalcase(TM2)isthecasefromthe1,415thathasthehighestvalueofthemetricforkeff=10.Thethird(TM3)andforth(TM4)trimodal 159

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casesarethecasesfromthe1,415thathavethehighestandlowestvalueofthemetricFforkeff=10,respectively. Table6-1. Parametervaluesofthesixcasesstudied.SMstandsforsinglemodal,BMforbimodal,andTMfortrimodal. Var.SMBMTM1TM2TM3TM4 A10.141420.100000.081650.127330.039150.00872A2-0.100000.081650.038920.116200.11528A3--0.081650.047670.070470.08145k110778114k2-11917122k3--131582510.000000.000000.000000.000000.000000.000002-0.000000.000002.052266.000173.923283--0.000004.841470.378183.13525keff9.999959.000009.666759.4767710.164129.67411 Figure 6-2 showsPVforthesixconsideredcases.Inthesecases,thedistancefromtheoriginquantiesthePVinthatsector.Pleasenotethatthecontoursdonotrepresentparticleposition.AgeneralobservationthatcanbemadebylookingatFig. 6-2 isthatparticlesmigratefromthesectorswithinitiallylowerPVtothesectorswithinitiallyhigherPV. 160

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Figure6-2. PVcontourevolutionforeachofthesixcasesconsideredinTable 6-1 161

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Althoughthetwometricswerebuilttomeasurethesameproperty,theamplicationofdeparturefromaxisymmetry,thepredictioninsomecasesisquitedierent.Themetricspresentacorrelationof0.77.ThecorrelationcoecientwascalculatedusingPearson'sLinearCorrelation[ 281 ].Thecorrelationbetweenthemissignicantalthoughnotoverwhelming(resultsbasedon1,415simulations)thereforeIexpectthattherewillbesomedierencesbetweenthemetricpredictions.Figure 6-3 showsFandasafunctionoftimeforthesixcasesconsidered.At500s,themetricsareconsistentforTM1,TM2,andTM4giventhethird-highest,thesecondlowestandthelowestvalues,respectively.SMisthesecondhighestpredictionforthemetricFandthehighestforthemetric.Ontheotherhand,TM3at500sgivesthehighestvalueofF,however,itgivesamediumvalueof.InFigure 6-3 wecanalsoseethatthemetricFhasasmoothergrowth.Thiscanbeeasilyexplainedasfollows,FisnormalizedbyA21+A22+A23whichhappenstobeaconstantbyconstruction.Ontheotherhand,thenormalizationofisdoneusingthedierencebetweenthemaximumandtheminimumvolumeofparticlesinthesectorsattheinitialtime.Thisdierencechangesfromcasetocase.Inotherwords,thevalueof~F(t=0)isequalto0.02(Eq.( 6-4 ))forallcases,whilethevalueofthenormalizationfactorPV(t=0)(Eq.( 6-5 ))isdierentforeachcasepenalizingthecaseswiththelargerinitialPVFdierencesbetweensectors.Figures 6-4 and 6-5 helptounderstandwhythecaseTM3givesthehighestdeparturefromaxisymmetryifwerelyonF,butamediumdepartureifwecompute.LetusrstobserveFigure 6-4 andFigure 6-5 .Figure 6-4 correspondstothecaseSMwhichhasaconsistentlyhighvalueforthetwometrics.Figure 6-5 correspondstothecaseTM3.Figure 6-4A andFigure 6-5A correspondtoinitialtimewhileFigure 6-4B andFigure 6-5B correspondtonaltime(500s).Asyoucansee,thedierenceinPVattheinitialtimeis50%higherinTM3thaninSM,whilethedierenceatalatertimeisonlyan8%higher.Therefore,TM3ispenalizedforstartingwithalargerPVdierenceandwasnotabletogrowaccordingly.NotethatforSM,PV(500s)increased362% 162

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A BFigure6-3. asafunctionoftimeforthesixcasesdescribedinTable 6-1 .A)FasafunctionoftimeforthesixcasesdescribedinTable 6-1 ;B)asafunctionoftimeforthesixcasesdescribedinTable 6-1 (Eq.( 6-7 ))incomparisonwithitsinitialvalue.Ontheotherhand,forTM3(Eq.( 6-8 ))itincreasedonlya232%. A BFigure6-4. ParticlevolumeasafunctionoftheangularcoordinateforthecaseSM.A)Initialtime;B)Finaltime. (500s)=0:72010)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(3m3 0:15610)]TJ /F1 11.955 Tf 7.08 -3.45 Td[(3m3=4:62(6-7) 163

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A BFigure6-5. ParticlevolumeasafunctionoftheangularcoordinateforthecaseTM3.A)Initialtime;B)Finaltime. (500s)=0:78410)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(3m3 0:23610)]TJ /F1 11.955 Tf 7.08 -3.45 Td[(3m3=3:32(6-8)Figure 6-6 showsthenormalizedspectrumsquaredforthecaseSMwhileFigure 6-7 forthecaseTM3.Figure 6-6A andFigure 6-7A correspondtoinitialtimewhileFigure 6-6B andFigure 6-7B correspondtonaltime(500s).IntheTM3case,theinitialmodescombinetogeneratenewmodesandtheiramplitudesgrowenoughtoreachthesinglemodeamplitude.Thisisinterestingbyitselfbecausethereisalossofenergyintheprocessofcreatingthesenewmodes,however,forTM3theamplitudeswereabletogrowandovertakethesinglemodalSMcase. 164

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A BFigure6-6. SquaredFourierspectranormalizedforthecaseSM.ThevalueofFatnaltimeis7.03.A)Initialtime;B)Finaltime. A BFigure6-7. SquaredFourierspectranormalizedforthecaseTM3.ThevalueofFatnaltimeis7.27.A)Initialtime;B)Finaltime. 165

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CHAPTER7MULTI-FIDELITYSURROGATESInthischapter,low-delity(LF)andhigh-delity(HF)simulationsareusedtoconstructmulti-delity(MF)surrogates,studytheirperformanceandcomparethemwiththeirsingle-delitycounterparts.Inthischapterdatapointsrefertotheset(x;y(x)),wherexisthevectorofdesignvariablesandy(x)isthesimulationresponsegiventheinputx.Notethatboth,variablesandresponse,areneededtobuiltsurrogates.^y(x)willrefertothesurrogateresponseatthevariablesx.Inthiswork,trainingdatapointsarethedatapointsusedtobuildasurrogate,whilevalidationdatapointsarethedatapointsusedtotestthesurrogateperformance.ThedetailsofthesimulationscanbefoundinChapter 4 ,Section 4.3 .TheLFdatapointsareobtainedfromsimulationsusingthegriddescribedinSubsection 4.3.2 .Thecomputationaldomainisdividedintotworegions,aninnerCartesianmeshwith6464cellswhichinitiallycontainsthehigh-pressuregas,andanouterannulusthatusesapolarmeshwith125and256cellsintheradialandazimuthaldirections,respectively.ThenumberofcomputationalparticlesinLFsimulationsis31;250.Particlesarerandomlydistributedwithintheannularregionwithuniformprobability.TheLFdatacosttwocorehoursinQuartzLawrenceLivermoreNationalLaboratoryhigh-performancecomputer.TheHFdatapointsareobtainedfromsimulationswitha16timesnergrid,256256cellCartesianmeshfortheinnergrid,andfortheoutergrida512and1024cellsintheradialandazimuthaldirections,respectively.Notethatthenumberofcomputationalparticleshasbeenincreasedaccordinglytomaintainthesamenumberofcomputationalparticlespercell,inthiscase,125;000.ThecomputationalcostoftheHFis56corehoursinQuartz.ThereforetheLFtoHFcostratiois4%. 7.1TheMulti-delitySurrogatesUsedAsdiscussedinChapter 4 ,Section 4.7.2 ,ourmetricspresentnoise,thereforetheLFandHFsurrogateswereconstructedusingtheclassicallinearregressionapproach[ 282 ] 166

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usingasbasisfunctionsmonomialsuptoasecondorderpolynomial.Regressionwaschosenduetoitsgoodperformancelteringthenoise.Otherssurrogates,likeKriging,werealsotestedbuttheresultsobtainedwerenotsatisfactory.BecauseofthefactthatKriginginterpolatesthetrainingdatapoints,itsperformanceinnoisyproblemscanbepoor,alsothehighnumberoftrainingdatapointsusedforsurrogateconstructionledKrigingtolousytting.TheMFsurrogateswerebuiltalsousingtheregressionapproachbutcombiningmorethanonedelity.FortheMFapproximationstheperformanceofsixdierentapproacheswerestudiedwhichhavebeendiscussedindetailinSection 3.4 .Nevertheless,theyarealsodiscussedbrieybelow.Givenalow-delitymodel,yLF(x),andahigh-delitymodel,yHF(x),theirsurrogatesaredenotedas^yLF(x)and^yHF(x),respectively.TheadditivecorrectionapproachassumesthattherelationbetweenyLF(x)andyHF(x)is ^yMF1(x)=^yLF(x)+^(x);(7-1)where^yLFisthelinearregressionsingle-delitysurrogatebuiltusingthe1,415LFdatapoints,and^isthesurrogateconstructedusingastrainingdatapointsthedierencebetweentheyHFandyLFfunctionsatthenestedtrainingdatapoints(inthiscaseupto711trainingdatapoints).Inotherwords^isthesurrogateof,thediscrepancyfunctionbetweenyHFandyLF.Nesteddatapointsarethosecomputedusingthesamevariablesbutusingdierentmodels.Themultiplicativeapproachis ^yMF2(x)=^(x)^yLF(x);(7-2)where^(x)isthesurrogateconstructedusingastrainingpointsthequotientbetweenyHF(x)andyLF(x)functionsatthenestedtrainingdatapoints(inthiscaseupto711datapoints).Thecomprehensiveapproachis ^yMF3=^yLF(x)+^(x);(7-3) 167

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wherethemultiplicativecorrectionisaconstant,thatis^(x)==constant.ThecomprehensivesurrogatewasconstructedusingLR-MFS[ 283 ],whichbasicallyconsistsofaddingtotheHFsurrogate,^yHF,constructedusingclassicallinearregressionanextrabasisfunctionthatdependsontheLFmodelevaluatedattheHFtrainingdatapoints.TheMFsurrogateswerealsoconstructedusingasecond-degreepolynomial.AllthemodelswerebuiltusingMATLABregressfunction.TheMFsurrogatepresentedinEq.( 7-3 )canbealsoapproximatedusingco-Kriging,wherethelikelihoodfunctionismaximized,insteadofminimizingtherootmeansquareerror(RMSE).AnadvantageoftheLR-MFSapproachisthatthe^yLF(x)canbeeasilyreplacedbyyLF(x)iftheLFmodelischeapenough.However,thisoptionisnotusuallyavailableforco-KrigingduetothefactthatasurrogateisconstructedinternallybasedontheLFtrainingdatapoints.Takingadvantageofthis,thethreeapproachespresentedabovewherealsoconstructedusingyLF(x)insteadof^yLF(x).Thatis,fortheadditivecorrection,Eq( 7-1 )willbecome ^yMF4(x)=yLF(x)+^(x);(7-4)where^(x)isthesurrogateconstructedusingastrainingdatapointsthedierencebetweentheyHF(x)andyLF(x)functionsatthenestedtrainingdatapoints(inthiscaseupto711points).Forthemultiplicativeapproach,therelationshipbetweentheHFandLFfunctionsisassumedtobe ^yMF5(x)=^(x)yLF(x);(7-5)where^(x)isthesurrogateconstructedusingastrainingdatapointsthequotientbetweenyHF(x)andyLF(x)functionsatthenestedtrainingdatapoints(inthiscaseupto711points).Forthecomprehensiveapproach,additiveandmultiplicativecorrectionsarecombined, ^yMF6=yLF(x)+^(x);(7-6)whereisaconstant. 168

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7.2TheDesignofExperimentsInSection 4.7.3 itwasshownthatfromtheoriginalninevariables,thethreevariablescorrespondingtotherelativephasebetweenmodeswerenotasinuentialastheothersix.DuetotheenergyconstraintshowninEq. 4-21 ,thevariableA3wasalsoignored.ThereforethevevariablesconsideredinthisstudyareA1,A2,k1,k2;andk3.Themainobjectiveofthisworkistomeasuretheamplicationofthedeparturefromaxisymmetry.Forthispurpose,twometrics,Fand,werecomputedwhichweredescribedinChapter 6 .ThesurrogatesforbothmetricswereconstructedwiththevariablesA1,A2,k1,k2,andk3usingupto1,415LFsimulationsand711HFsimulations.Knowingthecorrelationbetweentheavailabledatapointsgivesusanideaofwhatitisexpectedfromthesurrogateperformance.FromahighcorrelationbetweentheHFdatapointsandtheLFdatapointsitisexpectedagoodMFsurrogateperformancewhileotherwise,usingMFsurrogatescanbenotonlypoorbutharmful.Thatis,withlowcorrelation,theMFsurrogatemaybelessaccuratethanasurrogatebuiltusingonlyHFtrainingdatapoints.ThecorrelationcoecientbetweentheLFdatapointsandtheHFdatapointsforthemetricFis0:92whileforthemetricis0:71.Forcompleteness,thecorrelationcoecientbetweenthemetricswascalculated.Bothmeasuretheamplicationofthedeparturefromaxisymmetry,therefore,ahighcorrelationisexpected.ThecorrelationbetweentheLFdatapointsforthemetricFandtheLFdatapointsforthemetricis0:77.ThecorrelationbetweentheHFdatapointsforthemetricFandtheHFdatapointsforthemetricis0:67.ThisinformationissummarizedinTable 7-1 Table7-1. Correlationcoecientbetweenthedatapoints DataPointsComparedCorrelationCoecient LFvs.HF,metricF0.92LFvs.HF,metric0.71Fvs.,LFdata0.77Fvs.,HFdata0.67 169

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ThealgorithmusedtoobtainthedatapointsisanesteddesignofexperimentsbasedonLHStechnique[ 284 ].Theboundsinthevariablesare0
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A B C DFigure7-1. Themetrics,Fandasafunctionofkeff.LeftguresshowtheLFdatapoints,whilerightguresshowtheHFdatapoints.Thebluedotsarethetrainingdatapointswhiletheredcrossesarethevalidationdatapoints.A)LFFasafunctionofkeff;B)HFFasafunctionofkeff;C)LFasafunctionofkeff;D)HFasafunctionofkeff. TheHFsurrogateisonlytrainedwithHFdatapointsand,asthegureshows,asthenumberoftrainingpointsincreasestheperformanceimproves.ThecomprehensiveMFsurrogate^yMF6(Eq.( 7-6 ))usesasmanyLFdatapointsasHFdatapoints,therefore,asthenumberofHFdatapointsincreases,thenumberofLFdatapointsalsoincreases.ForthemetricF(Fig. 7-2A ),theMFsurrogatesthatusetheLFsurrogate,^yMF1^yMF2and^yMF3,haveaperformancesimilartotheHFsurrogate^yHF.ForF,theadditive,multiplicative,andcomprehensiveMFsurrogates^yMF4,^yMF5 171

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and^yMF6haveathreetimeshigherperformance.Thisperformanceisroughly5%.Thispredictioncannotbeimprovedduetothefactthatthenoiselevelinthemetricisofthatorder.Forthemetric(Fig. 7-2B )usingyLFor^yLFintheconstructionoftheMFsurrogatesdoesnotseemstomakeasubstantialdierence.Forbothmetrics,thebestperformanceisachievedbythecomprehensiveMFsurrogate^yMF6builtusingyLF.ForthemetricFthelowesterrorachievedis5%whileforthemetricis7%.Foralltheapproximations,afteradding100HFdatapoints,theperformancedoesnotchangesubstantially.Thisisbecause,besidesregression,basisfunctionuptoaquadraticpolynomialwerechosen.Ifthechosenfunctionwouldhavebeenmorecomplexortheorderofthepolynomialwouldhavebeenhigher,thenumberofpointsrequireduntilreachingaplateauwouldhavebeenhigher. A BFigure7-2. RMSEforbothmetrics,Fand,asafunctionofthenumberofHFdatapointsusedtotrainthesurrogatesdescribedinSection 7.1 .The100validationdatapointsusedtocomputetheerrorsaretheonesshowedinFigure 7-1 .A)F;B). InTable 7-2 andTable 7-3 therelativeRMSEforthemetricFand,receptivelyusingthemaximumnumberofHFdatapointsavailable(711)isshown.NotethatLFsurrogateRMSEdoesnotdependonthenumberofHFpoints.Thecoecientof 172

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determination,R2,wasaddedtothetableshaveanideaoftheproportionofthevarianceofthedependentvariablethatispredictedbytheindependentvariables.NotethatthecoecientwasalsocomputedforthecasethatusesthemaximumnumberofHFdatapoints. Table7-2. RelativeRMSEandcoecientofdetermination(R2)forthemetricFusing711HFdatapoints.Mult.,Add.,Comp.,Surr.,andSim.standforMultiplicative,Additive,Comprehensive,Surrogate,andSimulation,respectively SurrogateRel.RMSE(F)R2(F) LF0.2210.78HF0.1470.62Mult.Corr.(LFSurr.)0.1430.66Add.Corr.(LFSurr.)0.1460.50Comp.Corr.(LFSur.)0.1460.92Mult.Corr.(LFSim.)0.0720.66Add.Corr.(LFSim.)0.0590.50Comp.Corr.(LFSim.)0.0560.92 Table7-3. RelativeRMSEandcoecientofdetermination(R2)forthemetricusing711HFdatapoints.Mult.,Add.,Comp.,andSurr.standforMultiplicative,Additive,Comprehensive,andSurrogate,respectively SurrogateRel.RMSE()R2() LF0.1450.48HF0.090.62Mult.Corr.(LFSurr.)0.090.14Add.Corr.(LFSurr.)0.090.16Comp.Corr.(LFSurr.)0.090.7Mult.Corr.(LFData)0.090.14Add.Corr.(LFData)0.080.16Comp.Corr.(LFData)0.070.7 NowthecontributionoftheLFdatapointstotheMFsurrogateswillbestudiedinordertounderstandwhytheperformanceofthesurrogatesthatuseyLFinsteadoftheLFsurrogate^yLFforpredictionworkedoverwhelminglybetterforthemetricFbutnotforthemetric.First,noticethatEq.( 7-3 )canbeexpressedas yMF=^yLF(x)+pX1=1Xi(x)bi;(7-7) 173

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whereXi(x)istheithmonomialbasisandbiisthecoecientofXi(x).Similarly,Eq.( 7-6 )canbewrittenas, yMF=yLF(x)+pX1=1Xi(x)bi:(7-8)Inthiscasep=21,whichisthenumberofcoecientsofaquadraticpolynomialinvevariables.Also,notethatforpredictionwecanchoosebetweenusingyLFor^yLF,howeverfortrainingpurposesyLFwasused,thereforeandbiarethesameinEq.( 7-7 )andinEq.( 7-8 ).Equations( 7-7 )and( 7-8 )showexplicitlythecontributionofeachmodel,HFandLF,tothecomprehensiveMFapproximationusingthe^yLFandyLF,respectively.ThersttermrepresentsthecontributionoftheLFmodel,^yLF(x)(usingLFsurrogate),andyLF(x)(usingLFdatapoints)totheMFsurrogate.Thesecondterm,Pp1=1Xi(x)bi,representsthecontributionoftheHFmodeltotheMFsurrogate.InordertostudythecontributionoftheLFdatapointstotheMFsurrogatesasafunctionoftheHFdatapointsisplottedinFigure 7-3 .NotethatcanbeseenastheweightoftheLFdatapointsinthestructureofthecomprehensiveMFsurrogate.Alongwith,itisalsoplottedtherootsumofthecoecientsbisquared,thatis,p Ppi=1b2i.Figure 7-3A showsthat,forthemetricF,after200HFpoints,valueistentimeshigherthanthesquaredsumoftherestofthecoecientsinvolvedintheMFsurrogatestructure.However,forthemetric,thissummationisatleastthreetimeshigherthroughouttheentirerange(Fig. 7-3B ).ThisoutcomecouldbeexpectedduetothehighercorrelationbetweenLFandHFforFshowedinTable 7-1 .Figure 7-4 showsthemeancontributioninpercentageoftheLFandHFinformationtothecomprehensiveMFsurrogatepredictedusingyLF.UsingyLFor^yLFforpredictingtheMFcomprehensivecorrectiongivesthesameLFandHFmeancontributions,therefore,onlyoneplotwasincluded.Figure 7-4A showsthecontributionforthemetricF,whileFigure 7-4B forthemetric.Themeanwascalculatedaveragingthecontributionofthe100validationpoints.Anegativepercentagemeans,inthiscase,thatthecontributionhasnegativesignwhileapercentagehigherthan100%means 174

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A BFigure7-3. TheconstantandthesquaredsumofthecoecientsbiasafunctionoftheHFdatapointsforthecomprehensiveMFcorrectionsshowninFigure 7-2 .A)F;B) thatthemeancontributionishigherthanthemetricvalue.Naturally,thesumoftheLFandHFcontributionsaddsto100%.ForahighnumberofHFdatapoints,thecontributionisdominatedbytheLFinformationforthemetricF(85%)whileforthemetricisdominatedbytheHFinformation(65%).ThisalsohelpstoexplainwhatitwasobservedinFigures 7-2A and 7-2B .Thatis,formetricFtheresultsoftheMFcomprehensivepredictionusingyLFperformsoverwhelminglybetterthantheonesthatusetheLFsurrogate.WhenthesourceofLFischangedforprediction,theresultschangedrastically.NotethatevenifforourproblemtheMFsurrogatepredictionsusingyLFperformedbetterthanusingtheLFsurrogate,itcanbethecasethatusingtheLFsurrogateperformsbetterthanusingyLF.ForthemetricthecontributionoftheLFmodelissmaller,therefore,usingyLFinsteadoftheLFsurrogatedoesimprovetheprediction,however,theimprovementisnotoverwhelming.Figures 7-3 and 7-4 helptoexplainwhyachangeinthewaythattheLFdataisintroducedinthecomprehensiveMFsurrogatewhilepredicting,aectssubstantiallytheapproximationforFbutnotasmuchtheapproximationfor.Theotherreason 175

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A BFigure7-4. MeancontributionoftheLF(^yLF(x)fromEq.( 7-8 ))andoftheHF(Pp1=1Xi(x)bifromEq.( 7-8 ))modelstothecomprehensiveMFsurrogateprediction(usingLFdatapoints)inpercentageasafunctionoftheHFdatapointsused.A)F;B) isthattheLFsurrogateisaquadraticpolynomial,thatwhenisusedintheMFapproximations,alsotrainedusingaquadraticpolynomial,givesasaresponseanotherquadraticpolynomial.Therefore,partofthedispersionpresentinthefunctionthatwearetryingtopredictismissing.Ontheotherhand,ifweusedyLF,thegoodcorrelationbetweenLFandHFdatapointsallowsasubstantiallybetterprediction. 7.4UsingSymmetriesintheConstructionofMulti-delitySurrogatesInChapter 5 itwasshownthatourproblempresentsparametricsymmetries.AfteridentifyingthattheMFcomprehensivesurrogateusingyLFforpredictionworksthebestforbothmetrics,wegofurtherandweimposesymmetriestoimproveitsperformance.InFigure 7-5 arepresentedtheresultsofaddingpermutationpoints.ThiswasdoneusingtheapproachSBFdescribedinSection 5.4 whichmodiesthesurrogatebasisfunctionsinordertoaddthepermutationpointinformationinthesurrogate.WhileforahighnumberofHFdatapoints(100HFdatapoints)theperformance,usingandnotusingpermutationpoints,doesnotvarysubstantially,forlessthan60HFdatapointstheimprovementissubstantialforbothmetrics.ForthemetricF,usingpermutationpoints 176

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allowsachievingalmostthenalperformance(5%)usingtheminimumpossiblenumberofHFpoints(22pointsinthiscase).Thereforethenumberofpointsneededforthesameaccuracyhasreducedafactorofthree.Forthemetricthenalperformanceisachievedusingonly30HFdatapoints,whichreducesthecostbyhalf. A BFigure7-5. RMSEforbothmetrics,Fand,asafunctionofthenumberofHFdatapointsusedtotrainthesurrogatesusingpermutationpoints(continuousline)andnotusingpermutationpoints(dashedline)fortheMFsurrogate^yMF6.The100validationdatapointsusedaretheonesshowedinFigure 7-1 .A)F;B). Toconclude,inthecurrentproblemusingaquadraticpolynomialLR-MFSasMFsurrogate,predictingtheresponseusingyLFinsteadoftheLFsurrogate,givesthebestpredictionforbothmetrics.UsingthisMFsurrogate,theRMSEoftheHFsurrogateforthemetricFwasreduceda90%,improvingthetotalrelativeaccuracyfrom87%to95%.ForthemetrictheRMSEwasreduceda15%,improvingthetotalrelativeaccuracyfrom90%to93%.Furthermore,usingpermutationpointsthenalaccuracyisachievedusinglessthan30HFdatapoints,thisreducesthesimulationcostuptothreetimes. 177

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CHAPTER8CONCLUSIONSInthiswork,theevolutionofsmallperturbationsintheinitialparticlevolumefractionofparticledistributionwithinanannularbedsurroundingahigh-energyexplosiveinacylindricaltwo-dimensionalmultiphaseexplosionwasinvestigated.Theparticlevolumefractioniscomputedasparticlevolumeovercomputationalcellvolume.Theinitialperturbationsweresinusoidalintheangularcoordinateandconstantintheradialcoordinate.Iusedcombinationsofuptothreemodeswhilevaryingtheirparameters(amplitude,wavenumber,andrelativephase).Iobservethattheradialsectorswithlargerparticlevolumeappearedasporousnozzleswithalargerareacontraction,whilesectorswithsmallerparticlevolumeappearedasporousnozzleswithasmallerareacontraction.Thepost-shockowinthepresentproblemwassupersonic,thereforethenozzlingeectwastodecreasethevelocityinregionsoflargerparticlevolume.Thisreductionwasfurtherstrengthenedbytheincreaseddragonthegasowduetothehigherconcentrationofparticlesinthehighparticlevolumesectors.Theparticleslocatedinthehigh-speedsectorsalsogainedmoremomentumthanparticleslocatedinradialsectorswithhigherparticlevolume.Astheparticlesinsectorswithlowparticlevolumemovedfartheroutradially,theyalsotendedtocircumferentiallymigrateintotheslowmovinghighparticlevolumesectors.Thus,sectorsofhigherparticlevolumetendedtofurtherincreaseattheexpenseoflowparticlevolumesectors.Thisfeedbackmechanismwasthesourceofthechannelinginstabilityobservedinthepresentsimulations.Theneteectofthechannelinginstabilitywastoincreasetheangularvariationinthenetvolumeofparticlescontainedwithinradialsectorsofthedomainandalsotoincreasethedierenceintheradialextentofparticleswithinthedierentsectors.Boththesemechanismsappearedasparticledeparturefromaxisymmetry.Isoughttomeasure 178

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theamplicationoftheseinstabilitiesandthedeparturefromaxisymmetryintheparticledistribution.Idenedtwometrics,Fand,inordertoobtaininformationregardinghowtheparticlesaredistributedatlatertimesinthesimulations.Bothmetricsintendtomeasuretheamplicationoftheparticledeparturefromaxisymmetry.Fisametricthatdependsonthetotalamplitudeofallthecircumferentialmodespresent,includingboththeoriginalperturbationmodesandtheemergingnewones.isametricthatdependsonlyonthedierencebetweenradialsectorswithmaximumandminimumparticlevolume.Iobservedthattherelativephasesbetweenthemodesoftheinitialperturbationdonotplayanimportantroleinthemetricswhencomparedwiththeotherparameters(amplitudesandwavenumbers)and,furthermore,theirinuenceinthemetricsisontheorderofthenoise.Ialsofoundthattheeectivewavenumber,avariableconstructedasthesumofthewavenumbersweightedwiththesquaredamplitudes,tohavethestrongestinuenceonthemetrics,morethananyothervariable.Bothmetrics,Fand,aremaximizedatnaltime(500s)byinitialunimodalperturbations.Thisledtotheconclusionthatthenetsquaredamplitudesoftheemergingmodesinparticlevolumeandthemaximumdierencebetweenthenumberofparticlesinaradialsectoratthenaltimerelativetotheinitialdierence,ishighestifunimodalperturbationsareusedratherthanbimodalortrimodalperturbations.TheunimodaldisturbancethatmostampliedthemetricsFandarek1=20andk1=21,respectively.Butthewavenumberofthispeakmodecanbeexpectedtobedependentonthecircumferentialresolution.IconstructedsurrogatestopredictthemetricsFandthatmeasuretheamplicationofthedeparturefromaxisymmetry.Iusedlow-delity(LF)andhigh-delity(HF)simulationstoconstructLF,HF,andmulti-delity(MF)surrogates.Istudiedtheirperformanceandcomparethemwiththeirsingle-delitycounterparts.Linearregression 179

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withmonomialbasisfunctionuptoaquadraticpolynomialwasused.IfoundthatforthiscasetheMFmodelwasthemostaccurateforbothmetrics.TheMFsurrogatemodelsachievedaperformanceof5%forFand7%for.Thenoiseinthemetricsisroughly5%,thereforemoreaccuracyisnotpossible.Takingadvantageoftheparametricsymmetriesavailableinourproblem,Ialsomodiedthesurrogatemodelbasisfunctionstoimposethesymmetries.IcomparedMFsurrogatesbuiltwithandwithouttakingintoaccountsymmetries.IfoundthatifIusesymmetriesthenalaccuracyisachievedusing30HFdatapoints,insteadof100HFdatapoints,reducingthesimulationcostroughlythreetimes. 180

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BIOGRAPHICALSKETCHMaraGiselleFernandezwasborninCordoba,Argentinawhereshesuccessfullycompletedprimaryandsecondaryeducation.Shereceivedherbachelor'sdegreeinnuclearengineeringfromBalseiroInstitute,SanCarlosdeBariloche,RoNegro,ArgentinainJune2014.HerthesiswasfocusedonthesurveillanceprogramfortheArgentinianreactorCAREM25.SheappliedforgraduatestudiesattheUniversityofFlorida,Gainesville,FloridaandwasacceptedtothePh.D.program.Shereceivedhermaster'sdegreeinmechanicalengineering,summacumlaude,fromtheUniversityofFloridainDecember2016.ShereceivedherPh.D.inaerospaceengineering,summacumlaude,fromtheUniversityofFloridainDecember2018.ShewasunderthesupervisionofDr.RaphaelT.HaftkaandDr.S.Balachandar.SheiscurrentlyapostdoctoralresearcherattheCenterforCompressibleMultiphaseTurbulenceatUniversityofFloridafundedbytheU.S.DepartmentofEnergy,NationalNuclearSecurityAdministration,AdvancedSimulationandComputingProgram,asaCooperativeAgreementunderthePredictiveScienceAcademicAllianceProgram. 205